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\begin{document}

 
\title[Interpolated Ap\'ery numbers]{Interpolated Ap\'ery numbers,
  quasiperiods \\ of modular forms, and motivic gamma functions}

\author{Vasily Golyshev}
\address{Algebra and Number Theory Laboratory,
Institute for Information Transmission Problems,
Bolshoi Karetny 19, Moscow 127994, Russia}
\email{golyshev@mccme.ru}

\author{Don Zagier}
\address{Max Planck Institut f\"ur Mathematik,
Vivatsgasse 7, 53111 Bonn, Germany
and International Centre for Theoretical Physics,
Via Miramare, Trieste, Italy}
\email{dbz@mpim-bonn.mpg.de}

\subjclass[2000]{Primary }
\keywords{}

\date{}

\begin{abstract}
We extend the definitions of the sequences used by Ap\'ery in his proof of the irrationality
of~$\zeta(3)$ to non-integral values of the index and relate the value with index~$-1/2$ to the central value of
the $L$-series of the unique normalized cusp form of weight~4 on~$\G_0(8)$. We also discuss the notion
of quasiperiods of modular forms and relate the Ap\'ery numbers of other half-integral indices to these. 
We further explain the conjectural relationship of the Taylor expansion around~0 of a different interpolation 
of the Ap\'ery numbers to a generalized version of the Gamma Conjecture, and discuss interpretations of 
the various results with families of Calabi-Yau manifolds, mirror symmetry, and motivic gamma functions.
\end{abstract}

\dedicatory{Dedicated to the memory of Boris Dubrovin, inspirational mathematician}


\maketitle


\section*{Introduction: the Ap\'ery numbers and their interpolations}

The Ap\'ery numbers  $A_0=1$, $A_1=5$, $A_2=73$,\,\dots\ are defined recursively by 
\begin{equation}\label{AJ}  (n+1)^3\,A_{n+1} \m (34n^3+51n^2+27n+5)\,A_n \+ n^3A_{n-1}\;=\;0 \end{equation}
together with the initial value~$A_0=1$ and an arbitrary value of~$A_{-1}$.  The fact that they are all integers, 
which is not at all obvious from this definition, played a key role in Ap\'ery's famous proof of the irrationality 
of~$\z(3)$ in~1978.  It can be understood from at least three different points of view, one hypergeometric, one modular, 
and one algebraic-geometric.  The first is due to Ap\'ery himself, who gave the explicit closed formula
\begin{equation}\label{AA}  A_n\;=\; \sum_{k=0}^n {\binom nk}^2\,{\binom{n+k}k}^2 \qquad\quad(n\ge0) \end{equation}
for his numbers. This makes their integrality obvious but does not really explain why it holds.  
A more conceptual explanation in terms of modular forms was found by Beukers~\cite{Beukers1987}: the generating series
\begin{equation} A(t) \;=\; \sum_{n=0}^\infty A_n\,t^n \;=\; 1 + 5t + 73t^2 + 1445t^3 +\,\cdots \end{equation}
of the Ap\'ery numbers, which satisfies a third-order differential equation corresponding to the recursion~\eqref{AJ}, 
can be parametrized by
\begin{equation}\label{ApMod} A(t_6(\tau)) \;=\; f_6(\tau), \end{equation}
where
\begin{equation} t_6(\tau) \;=\; \frac{\eta(\tau)^{12}\,\eta(6\tau)^{12}}{\eta(2\tau)^{12}\,\eta(3\tau)^{12}} \;=\; q - 12q^2 + 66q^3 - 220 q^4 +\,\cdots \end{equation}
($q=e^{2\pi i\tau}$, $\eta(\tau)\,=$ Dedekind eta-function) is a Hauptmodul for the group $\G_0(6)$ and 
\begin{equation} f_6(\tau) \;=\; \frac{\eta(2\tau)^7\eta(3\tau)^7}{\eta(\tau)^5\,\eta(6\tau)^5} \;=\; 1 + 5q + 13q^2 + 23q^3 + \,\cdots  \end{equation}
a certain modular form (Eisenstein series) of weight~2 on~$\G_0(6)$, and the integrality of the coefficients of~$A$ follows
from that of~$t_6$ and~$f_6$.  Finally, the algebraic-geometric explanation, which was given (slightly earlier) by Beukers and 
Peters~\cite{BeukersPeters1984}, identifies the differential equation satisfied by~$A(t)$ with the Picard-Fuchs differential
equation of the family of K3~surfaces given (up to birational equivalence) by the equation 
\begin{equation} \label{Laurent} \frac{(y-1)(z-1)(x+z-1)(yz-x-z-1)}{xyz} \;=\; \frac1t\,. \end{equation}
More explicitly, one checks that $A_n$ is the constant term of the $n$th power of the Laurent polynomial occurring on the 
left-hand side of~\eqref{Laurent} (so-called Landau-Ginzburg model), from which its integrality is again obvious.
Each of these aspects is discussed in detail in~\cite{Zagier2018} a long survey of the arithmetic and geometric properties 
of differential equations in which the Ap\'ery numbers were used as the running example to illustrate the theory and 
in which many of the results of the present paper were announced.

In this paper we will extend these ideas in several different directions, essentially by adding a variable 
and thinking of the original results as those obtained by ``fibering out" a larger motive over
a torus
and seeing how this affects the local or monodromy properties of the arithmetic of the individual fibers.
Specifically, we will see:

\smallskip $\bullet$ that there is a natural holomorphic extension of the Ap\'ery numbers to complex values 
of the argument~$n$, satisfying a modified form of the same recursion~\eqref{AJ} and symmetric with respect to~$n\mapsto-n-1$\,;

\smallskip $\bullet$ that the interpolated value of $A_n$ at the point of symmetry $n=-1/2$ is proportional
to the central value of the Hecke $L$-series of a certain cusp form~$\ff$ of weight~4 and level~8;

\smallskip $\bullet$ that the interpolated values of $A_n$ at other half-integral arguments of~$n$ are
related to the periods and quasiperiods (whose definition we will review) of the same form~$\ff\,$;

\smallskip $\bullet$ that similar statements are true for the second Ap\'ery-like sequence $\{B_n\}=\{1,8,88,\dots\}$
defined by the recurrence relation
\begin{equation} \label{AK}  (n+1)^3\,B_{n+1} \m 8(4n^3+6n^2+4n+1)\,B_n \+ 256n^3\,B_{n-1}\;=\;0\,, \end{equation}
with the interpolated values $B_{n-1/2}$ again related to the periods and quasiperiods of~$\ff$;

\smallskip $\bullet$ that these facts are related to the monodromy of the Picard-Fuchs differential 
equation of a certain family of 3-dimensional Calabi-Yau manifolds, one of 14 such families for which 
this differential equation is hypergeometric and for all of which a similar relationship was verified
numerically in~\cite{KlemmScheideggerZagier};

\smallskip $\bullet$ that there is a second, and in some ways more natural, extension of the Ap\'ery numbers 
(or the~$B_n$) to complex arguments based on the asymptotics of the solutions of the recursion~\eqref{AJ} 
(or~\eqref{AK}) at infinity, but now satisfying the same recursion as the original numbers without any modification; 
and finally, 

\smallskip $\bullet$ that the first Taylor coefficients at zero of this new interpolation have an interpretation
in terms 
of the gamma class. The recursion \eqref{AJ}
appears as the Mellin transform of the regularized quantum differential equation
of its plane in this context, see \cite{Golyshev2007}. This phenomenon was 
found experimentally in the earlier paper~\cite{GolyshevZagier2016} 
and is shown here to confirm the prediction of the so-called Gamma Conjecture
for a particular 10--dimensional Fano variety.
 
\smallskip In the rest of the paper we will discuss each of these aspects in turn.

 \smallskip\section{Hypergeometric interpolation, periods, and quasiperiods}

As stated above, our purpose in this paper is to extend the definition of the sequence $\{A_n\}$ to non-integral values 
of~$n$ in two different ways and show how each of them leads to interesting extensions of some of the properties of the 
original numbers. The first, and most straightforward, interpolation arises by observing that we can rewrite~\eqref{AA} 
for $n\in\Z_{\ge0}$ as
\begin{equation}\label{AAA}  A_n\;=\; \sum_{k=0}^\infty {\binom nk}^2\,{\binom{n+k}k}^2\,, \end{equation}
since $\binom nk$ vanishes for $k>n$. The formula now makes sense for any complex value of~$n$, and the function that it defines
does indeed interpolate the original Ap\'ery numbers~$A_n$, but somewhat surprisingly it satisfies only a modification of
the original recursion involving an additive correction term that vanishes at all integer arguments:
\begin{theorem}\label{ThmA1}
The series~\eqref{AAA} converges absolutely and locally uniformly for all~$n\in\C$ and defines a holomorphic function in
the entire complex plane. This function is symmetric under $n\mapsto -n-1$ and satisfies the  functional equation
\begin{equation}\label{AJJ}  (n+1)^3\,A_{n+1} \m (34n^3+51n^2+27n+5)\,A_n \+ n^3\,A_{n-1}\;=\; \frac 8{\pi^2}\,(2n+1)\,\sin^2\pi n\;.  \end{equation}
\end{theorem}
A more interesting fact, which is the point of departure for the current paper, is that the value of the interpolated function
$A_n$ at its symmetry point $n=-1/2$ is a simple multiple of the central value of the Hecke $L$-series of a certain cusp form: 
\begin{theorem}\label{ThmA2}
Define $A_n$ for $n\in\C$ by~\eqref{AAA}. Then 
\begin{equation} \label{AB}   A_{-1/2} \;=\; \frac{16}{\pi^2}\,L(\ff,2)\,,  \end{equation}
where 
\begin{equation}\label{MainEq} \ff(\tau) \;=\;\eta(2\tau)^4\eta(4\tau)^4 \;=\;  q - 4q^3 - 2q^5 + 24q^7 -\, \cdots \end{equation}
is the unique normalized Hecke eigenform in $S_4(\G_0(8))\,$.
\end{theorem}

The statement of Theorem~\ref{ThmA2} is attractive, but at the same time somewhat mysterious, since it relates~$A_{-1/2}$ to 
a modular form on a completely different modular group from the one occurring in the modular parametrization~\eqref{ApMod}.
The explanation was sketched in~\cite{Zagier2018} and has to do with the fact that the double cover of the variety~\eqref{Laurent}
obtained by replacing~$t^{-1}$ by $w^2$ has a Hasse-Weil zeta function containing $L(\ff,s)$ as a factor. But then
there is a second mystery. The cusp form~$\ff$, like any normalized Hecke eigenform, has two basic periods, which we can
take to be $\om_+$ and $i\om_-$ with $\om_\pm\in\R$ defined by
\begin{equation}
  \om_+ \,=\, \int_0^\infty \ff(it)t\,dt \,=\, \frac{L(\ff,2)}{4\pi^2}\,, \quad
\om_- \,=\, \int_0^\infty \ff(it) dt \,=\, \frac{L(\ff,1)}{2\pi}\,=\, \frac{2\,L(\ff,3)}{\pi^3}\;.  \label{f8periods}
\end{equation}
On the other hand, from~\eqref{AJJ} we see that the values of $A_n$ for $n\in\Z+\h$ lie in the $\Q$-vector space spanned
by the three numbers $A_{-1/2}$, $A_{1/2}$, and $\pi^{-2}$. Since we know from Theorem~2 that $A_{-1/2}=64\om_+$, 
it is reasonable to guess that $A_{1/2}$, and therefore also $A_{n+1/2}$ for any $n\in\Z$, is in the $\Q$-linear span of 
the three numbers $\om_+$, $\om_-$, and $\pi^{-2}$.  But this turns out not to be the case.  We can compute all of the
numbers involved to high precision (the series defining $A_{\pm1/2}$ and $L(\ff,3)$ converge very slowly and the one defining 
$L(\ff,2)$ diverges, but they can be all computed rapidly and accurately by using standard convergence acceleration techniques 
and the integral representation of the $L$-function).  When we do so, we indeed find
$$ A_{-1/2} \,=\,64\,\om_+ \,=\,1.1186363871641870683496192575256409167948575515294\cdots\;, $$
in accordance with Theorem~\ref{ThmA2}, but the two further numerical values 
 \begin{align*} A_{1/2} &\,=\, 1.6719543024114185394024524459306932856672407502037904790004\cdots\;, \\
 64\,\om_- &\,=\, 3.6109143132953392967615735950910519401218540144056782090237\cdots  \end{align*}
do not yield any $\Q$-linear dependencies among $A_{-1/2}$, $A_{1/2}$, $\pi^{-2}$, and $\om_-$ with small coefficients.

The resolution of this mystery lies in the notion of {\it quasiperiods} of modular forms.
These numbers, which can be viewed as ``algebraic de Rham theory made concrete," are a simple 
generalization of differentials of the second kind, extending the well-known corresponding notion for 
elliptic curves discovered by Legendre in the 18th century.  They were discovered in the context of
modular forms and their entire theory worked out by Martin Eichler~\cite{Eichler1957} in 1957, but were then 
forgotten and rediscovered some 60 years later by Francis Brown~\cite{Brown2019},~\cite{BrownHain2018} and
also by us at about the same time (Spring~2015) in the context of the present paper; 
they are also studied and calculated
numerically in~\cite{KlemmScheideggerZagier} in connection with the transition matrices of solutions of
the Picard-Fuchs differential equation for hypergeometric families of Calabi-Yau 3-folds. 
We will summarize 
their definition and main properties briefly in Section~4, referring to any of the above-cited papers for more details.  
Roughly speaking, to any Hecke eigenform~$f$ one can associate two quasiperiods, well-defined up to an algebraic 
(and in this case rational) multiple of its two periods, defined by integrating a certain meromorphic modular form
over appropriate cycles. Computing these two quasiperiods for~$\ff$ numerically to high precision then suggests
the following theorem, whose more precise statement and proof will be described briefly in Section~4 
and which will follow essentially the same lines as the proof of~Theorem~\ref{ThmA2} given in the next section:
\begin{theorem}\label{ThmA3} 
The number~$A_{1/2}$ defined by~\eqref{AAA} is a rational linear combination of $A_{-1/2}$, $1/\pi^2$,
and the real quasiperiod associated to the Hecke eigenform~$\ff$. 
\end{theorem}

\medskip
\section{Proof of Theorems 1 and~2} 

The proofs of both Theorems~\ref{ThmA1} and~\ref{ThmA2} were given in~\cite{Zagier2018} 
(and in fact, as mentioned there, an identity equivalent to~\eqref{AB} was proved independently at about the same time 
by Rogers, Wan, and Zucker~\cite{RogersWanZucker2015}), but we will repeat them briefly here for the reader's convenience and because 
the same arguments with only minor modifications work also for Theorem~3 as well as for the corresponding statements
for the $B$-sequence discussed in the next section.

 It is convenient to shift the index by~$\h$, since this puts the symmetry point of the function at~0 and also 
makes the functional equation simpler. We therefore define a function $\cA(x)$ with $\cA(n+\h)=A_n$ by
\begin{equation} \label{AAx}  \cA(x) \;=\; \sum_{k=0}^\infty \alpha_k(x),\qquad \alpha_k(x) \,=\, {\binom{x-1/2}k}^2{\binom{x+k-1/2}k}^2 \,. \end{equation}
The statements about absolute and locally uniform convergence follow easily from Stirling's formula and standard
properties of the gamma function, which show that $\alpha_k(x)=\text O(k^{-2})$ as $k\to\infty$, and the symmetry
under $x\mapsto-x$ (corresponding to $n\mapsto-n-1$ for the numbers~$A_n$) is obvious because $\alpha_k(x)$ can
be written as ${\binom{x-1/2}k}^2{\binom{-x-1/2}k}^2$. Finally, induction on~$K$ gives
$$ \sum_{k=0}^K \bigl((x+\h)^3\alpha_k(x+1) - (34\,x^3+\tfrac32x)\alpha_k(x)+(x-\h\bigr)^3 \alpha_k(x-1))
  \,=\, 8x\,(2K^2+K-4x^2)\,\alpha_K(x) $$
for all integers $K\ge0$, and by computing the limiting value of the right-hand side as~$K\to\infty$ by Stirling's
formula we find that
\begin{equation}\label{ShiftedRec} \bigl(x+\h\bigr)^3\cA(x+1) \m \bigl(34\,x^3+\tfrac32x\bigr)\cA(x)
\+\bigl(x-\h\bigr)^3 \cA(x-1)   \;=\; \frac{16}{\pi^2}\,x\,\cos^2\pi x\,, \end{equation}  %
which is equivalent to equation~\eqref{AJJ}. This completes the proof of Theorem~1.

For Theorem~2, we rewrite the hypergeometric series defining~$A_{-1/2}$ as an integral involving modular functions
and modular forms, but on a different group than the one occurring in the modular parametrization~\eqref{ApMod}.
Define a function $\bb(\la)$ in the closed unit disc by
  \begin{equation}\label{DefBeta} \bb(\la) \;=\; F\bigl(\h,\h;1;\la) \;=\; \sum_{k=0}^\infty {\binom{-1/2}k}^2\,\la^k \qquad(|\la|\le1)\,, \end{equation}
where $F(a,b;c;x)$ denotes the Euler-Gauss hypergeometric function.  Then we have
\begin{equation}\label{BetaInteg} \cA(0) \;=\; \sum_{k=0}^\infty {\binom{-1/2}k}^4 
  \;=\; \frac1{2\pi i}\oint_{|\la|=1} \bb(\la)\,\bb(1/\la)\,\frac{d\la}\la\;.\end{equation}
On the other hand, we have the well-known modular parametrization
 \begin{equation}\label{BetaPar} \bb(\la(\tau)) \;=\; \vth_3(\tau)^2\,, \end{equation}
where $\la(\tau)$ (Legendre function) is the standard Hauptmodul for $\G(2)$, defined by
  $$ \lambda(\tau)\;=\;16\frac{\eta(\tau/2)^8\eta(2\tau)^{16}}{\eta(\tau)^{24}}\;=\;1- \frac{\eta(\tau/2)^{16}\eta(2\tau)^8}{\eta(\tau)^{24}}
   \;=\; \biggl(\frac{\vth_2(\tau)}{\vth_3(\tau)}\biggr)^4 \,, $$
in terms of the two Jacobi functions of weight 1/2
$$ \vth_2({\tau}) \;=\; \sum_{n\in\Z}q^{(n+\frac12)^2/2}\;=\;2\,\frac{\eta(2\tau)^2}{\eta(\tau)}\,,\qquad
\vth_3({\tau}) \;=\;\sum_{n\in\Z}q^{n^2/2}\;=\;\frac{\eta(\tau)^5}{\eta(\tau/2)^2\,\eta(2\tau)^2}\,.  $$
Using the modular transformation properties
$$  \frac1{\la(\tau)} \;=\; \la\Bigl(\frac\tau{1-\tau}\Bigr)\,, \qquad \vth_3\Bigl(\frac\tau{1-\tau}\Bigr)^2\;=\;(1-\tau)\,\vth_2(\tau)^2 $$
and the modular form identity
$$ \frac1{2\pi i}\,\vth_3(\tau)^2\,\vth_2(\tau)^2\,\frac{\la'(\tau)}{\la(\tau)} \;=\; 2\,\ff(\tau/4)\,, $$
with $\ff$ defined as in~\eqref{MainEq}, we obtain the integral representation
\begin{equation}\label{cA0integral} \cA(0)  \;=\; 2\int_0^2 (1-\tau)\,\ff(\tau/4)\,d\tau\,, \end{equation}
where the integral is taken along the hyperbolic geodesic from~0 to~2 (= Euclidean semicircle with center~1 and radius~1),
which is mapped by~$\la$ isomorphically to the unit circle.  Since $\ff$ is a cusp form, we can replace this path of integration
by the difference of the two vertical lines from 0 to~$i\infty$ and from~2 to~$i\infty$, and since $\ff(\tau+\h)=-\ff(\tau)$ (because
$\ff$ has a $q$-expansion containing only odd powers of~$q$), this gives finally
$$ \cA(0) \;=\; 2\,\Bigl(\int_0^\infty-\int_2^\infty\Bigr)\,(1-\tau)\,\ff(\tau/4)\,d\tau
   \;=\; -4\,\int_0^\infty \!\tau\,\ff(\tau/4)\,d\tau \;=\; 64\,\om_+ $$
as desired.

 \smallskip

\section{A second Ap\'ery-like sequence and the proof of Theorem~3} 

To understand the situation better we will consider not only the Ap\'ery numbers~$A_n$, but also a second sequence
defined by the terminating hypergeometric sum
\begin{equation} \label{AD}  B_n\;=\; \sum_{k=0}^n {\binom{2k}k}^2\,{\binom{2n-2k}{n-k}}^2 \quad\qquad(n\ge0) \end{equation}
and satisfying the recurrence relation~\eqref{AK}.
We will give the modular interpretation of this sequence in a moment, but first we do the analogue of what was 
done in~\S1 by describing an interpolation of $B_n$ to non-integral values of~$n$. This has both similarities and 
points of diference with the $A_n$ case.

The most obvious way to interpolate the $B_n$, imitating what we did in the Ap\'ery case, would be to 
rewrite the sum in~\eqref{AD} in the form
 $$  B_n\;=\; {\binom{2n}n}^2\,\sum_{k=0}^\infty {\binom nk}^4\,{\binom{2n}{2k}}^{-2}  $$
and then take the right-hand side of this expression as a definition for all values of~$n$, where
the pre-factor ${\binom{2n}n}^2$ is interpreted as $\frac{\G(2n+1)^2}{\G(n+1)^4}$, since the sum again can
be checked to converge like~$1/k^2$.  However, the function that we would obtain this way 
would be only meromorphic, with double poles at all half-integers. 
Instead, we replace the sum in~\eqref{AD} by 
\begin{equation} \label{BBy}
 B_n\;=\;    16^n\,\sum_{k=0}^\infty {\binom {-\h}k}^2\biggl(\frac{\G(\h)}{\G(n-k+1)\G(k+\h-n)}\biggr)^2\,,  
\end{equation}   
which again agrees with the original definition if~$n$ is a non-negative integer but now makes sense
for any complex number~$n$, 
since the sum converges like~$1/k^2$ and has no poles. This gives a natural
holomorphic continuation of the function $n\mapsto B_n$ to the complex plane.  This extrapolation is actually
simpler than the one for the Ap\'ery numbers, because the value of $B_n$ for integral $n<0$ vanishes (as opposed
to $A_n$ for $n<0$, which coincides with $A_{|n|-1}$) and the three-term functional equation satisfied by~$B_n$
remains true unchanged for complex values of~$n$ (as opposed to~$A_n$, where the right-hand side of the identity
had to be multiplied by a multiple of $\sin^2\pi n$).
We state these results in the form of a theorem, again shifting the index by 1/2 to make the functional equation and 
other properties of the function simpler, and also removing the factor~$16^n$ in the above sum, which was useful
for integrality when $n$ was integral but is only a nuisance when it is not.  
\begin{theorem}
The sum
\begin{equation}\label{DefB} \cB(x) \;:=\; \sum_{k=0}^\infty {\binom {-\h}k}^2\biggl(\frac{\G(\h)}{\G(x-k+\h)\G(k+1-x)}\biggr)^2  \end{equation}
converges for all complex values of~$x$ and defines an entire function with $\cB(n+\h)=16^{-n}B_n$ for $n\in\Z_{\ge0}$,
$\cB(n+\h)=0$ for $n\in\Z_{<0}$ and $\cB(-n)=\cB(n)$ for~$n\in\Z$, and satisfying the functional equation
\begin{equation}\label{AE} \Bigl(x+\frac12\Bigr)^3\cB(x+1) \m \Bigl(2\,x^3+\frac x2\Bigr)\cB(x) \+\Bigl(x-\frac12\Bigr)^3 \cB(x-1) \;=\; 0\,. \end{equation}
for all~$x\in\C$. The values of~$\cB(x)$ and~$\cA(x)$ at $x=0$ and $x=1$ are related by the formulas
\begin{equation}\label{cB1andcA1} \cB(0)\;=\;\cA(0)\,, \qquad 6\,\cB(1) \;=\; \cA(0) \+ \cA(1) \+ \frac1{\pi^2}\;. \end{equation} 
\end{theorem}
The proof of this is very similar to that of Theorem~1, so we skip the details.  The $1/k^2$ convergence of the series~\eqref{DefB}
is checked as before and the interpolation properties  $\cB(n+\h)=B_n$ for $n\in\Z_{\ge0}$ and $\cB(n+\h)=0$ for $n\in\Z_{<0}$
are obvious, since the series defining $\cB(x)$ agrees term-for-term with that defining~$B_{x-1/2}$ if~$x$ is a positive half-integer
and vanishes term-for-term (actually doubly, so~$\cB$ has double zeros at these points) if~$x$ is a negative half-integer.
The symmetry property for $x\in\Z$, which was obvious for $\cA(x)$ because each term of the series defining it was individually even, 
now follows from the recursion~\eqref{AE}, which for~$x=0$ gives $\cB(1)=\cB(-1)$ and then for other integral values of~$x$ gives the
equality between~$\cB(x)$ and $\cB(-x)$ by induction.  The functional equation is proved just the same way as for the Ap\'ery numbers, 
with the sum obtained by inserting the definition into all three terms on the left telescoping.  Finally, the last
statement of the theorem follows by direct comparison of the hypergeometric sums defining the numbers in question:
at $x=0$ we have
$$ \cB(0) \;=\; \sum_{k=0}^\infty{\binom{-\frac12}k}^4 \;=\; \cA(0) $$ 
while at $x=1$ we have
$$ \cA(1) \;=\; \sum_{k=0}^\infty{\binom{\frac12}k}^2{\binom{-\frac32}k}^2\,,$$
and 
\begin{equation}\label{cB1}  \cB(1) \;=\; \sum_{k=1}^\infty {\binom{-\frac12}k}^2{\binom{-\frac12}{k-1}}^2 
    \;=\; \sum_{k=0}^\infty 4k^2\,{\binom{-\frac12}k}^2{\binom{\frac12}k}^2 \,, \end{equation}
so that the desired formula $\cA(0)+\cA(1)-6\cB(1)=\pi^{-2}$ follows as in the proof of Theorem~1 by taking the limit 
as~$K\to\infty$ of the inductively proved identity
$$ \sum_{k=0}^K \biggl[{\binom{-\frac12}k}^4 \+ {\binom{\frac12}k}^2{\binom{-\frac32}k}^2 - 24k^2\,{\binom{-\frac12}k}^2{\binom{\frac12}k}^2\biggr]
 = 2\,(2K+1)^2\,{\binom{-\frac12}K}^4.$$

\medskip
We end by giving the modular interpretation of the numbers~$B_n$, or rather of their generating series
$$ B(t) \;=\; \sum_{n=0}^\infty B_n\,t^n \;=\; 1 + 8q +  88q^2 + 1088q^3 +\,\cdots\,.$$
Since $\binom{2k}k=4^k\binom{-1/2}k$, we see that this series is simply $\bb(16t)^2$, with~$\bb(\la)$ as in~\eqref{DefBeta},
so the identity~\eqref{BetaPar} used in the proof of Theorem~2 immediately gives the modular parametrization 
 \begin{equation}\label{BMod} B(t_4(\tau)) \;=\; f_4(\tau) \end{equation}
of $B(t)$, where 
 \begin{equation}\label{DEFt4} t_4(\tau) \;=\; 16\la(2\tau) \;=\; \frac{\eta(\tau)^8\eta(4\tau)^{16}}{\eta(2\tau)^{24}} 
          \;=\; q-8q^2+44q^3-192q^4 + \cdots \end{equation}
is a Hauptmodul for~$\G_0(4)$ (because $\la(\tau)$ is a Hauptmodul for~$\G(2)$) and
 \begin{equation}\label{DEFf4} f_4(\tau) \;=\; \theta_3(2\tau)^2  \;=\; \frac{\eta(2\tau)^{20}}{\eta(\tau)^8\eta(4\tau)^8} 
  \;=\; 1 + 8q + 24q^2 + 32q^3 + 24q^4 + \cdots \end{equation}
is a modular form (Eisenstein series) of weight~2 on~$\G_0(4)$. All of this is exactly like the situation for the~$A$'s,
and leads to the same mystery, though now with different numbers: while there the interpolated values of a sequence 
having a modular parametrization coming from level~6 were related to an $L$-value of a cusp form~$\ff$ with the different 
level~8, here a sequence whose modular parametrization comes from level~4 is related (by virtue of the last line of 
Theorem~4) to the same level~8 $L$-value.  The underlying geometry for this will be discussed in Section~5, but first
we should describe the quasiperiods associated to the form~$\ff$ and their relationship to the numbers $A_{n+1/2}$
and~$B_{n+1/2}$. 
 
We refer the reader to the unpublished work of A. Kalmykov \cite{Kalmykov},
where the series $B(t)$ is studied in the context of mirror symmetry.  

\section{Periods and quasiperiods of modular forms}

In this section we review the definitions and basic properties of periods and quasiperiods of modular forms and then
briefly describe the calculation of the quasiperiods in the case of the Hecke eigenform~$\ff$ and the proof of Theorem~3.

 We begin first by recalling the definition of the periods of a Hecke eigenform.  For $\ff$ we could
define the two periods $\om_+$ and $i\om_-$ by equation~\eqref{f8periods}, but for other eigenforms~$f$
of weight~4 this would not always work since $L(f,2)$ might vanish, and for forms~$f$ of weights~$k$ smaller or
larger than~4 it is not the right approach at all since there are $k-1$ critical values $L(f,n)$ ($0<n<k$)
but only two independent periods. A better way is to use Eichler integrals.  Let $f$ be a Hecke eigenform
of weight~$k$ on a congruence subgroup~$\G$  of $SL_2(\Z)$ (in our cases, always $\G_0(N)$ for some~$N\in\N$).
We will assume for convenience that $f$ has Fourier coefficients in~$\Q$, and will also not worry about 
integrality conditions, defining only a 2-dimensional $\Q$-vector space of periods with basis $\om_+,\,i\om_-$
for some numbers $\om_\pm\in\R^\times/\Q^\times$. For any $f\in M_k(\G)$ we denote by $\wt f$ an Eichler integral 
of~$f$, i.e., any holomorphic function in the upper half-plane satisfying $D^{k-1}\wt f=f$, where 
$D=q\,\frac d{dq}=\frac1{2\pi i}\,\frac d{d\tau}$ is the normalized differentiation operator. Bol's identity 
tells us that $D^{k-1}\bigl(\wt f|_{2-k}g\bigr)=f|_kg$ for any $g\in SL_2(\R)$, where $|_kg$ denotes the usual ``slash" 
operator $f|_k\bigl(\smallmatrix a&b\\c&d\endsmallmatrix\bigr)\bigl(\tau\bigr)=(c\tau+d)^{-k}f\bigl(\frac{a\tau+b}{c\tau+d}\bigr)$,
and hence that $r_{f,\g}:=\wt f-\wt f|_{2-k}\g$ is a polynomial in~$\tau$ of degree $\le k-2$ for every~$\g\in\G$.
The map $r_f:\g\mapsto r_{f,\g}$ is then a cocycle on~$\G$ with values in the space $V=V_{k-2}$ of all such 
polynomials (with the $\G$-action $P\mapsto P|_{2-k}\g$),
and the induced map from $M_k(\G)$ to $H^1(\G,V)$ is injective.  If $f=\sum_{n=1}^\infty a_nq^n$ is a cusp 
form, we can normalize the choice of~$\wt f$, and hence of the cocycle $r_f$ within its cohomology class $[r_f]$, 
by choosing $\wt f(\tau)=\sum_{n=1}^\infty n^{1-k}a_n\,q^n$, and if $f$ is also a Hecke eigenform as above, then with
this choice all of the polynomials $r_f(\g)$ belong to $V(\Q)\om_+\oplus V(\Q)i\om_-$ for some 
real numbers $\om_\pm$ that are well-defined up to rational multiples. If $\G$ is $\G_0(N)$ and we enlarge it
by adding the Fricke involution $W_N=\frac1{\sqrt N}\bigl(\smallmatrix 0&{-1}\\N&0\endsmallmatrix\bigr)$, under
which $f$ is automatically invariant up to sign, then the specialization of this statement to $\g=W_N$ gives 
the usual proportionality of the critical values of $L(f,n)$ (up to powers of~$\pi$) with the numbers $\om_\pm$.

For the quasiperiods we have to refine this story.  The cocycle $r_f$ for $f\in S_k(\G)$ belongs 
to the subgroup $H^1_{\rm par}(\G,V)$ of parabolic cohomology (meaning that $r_f(\g)\in V |_{2-k} (1-\g) $ for all  
parabolic elements $\g$ of~$\G$), and the map $f\mapsto[r_f]$ together with its complex conjugate induce an isomorphism 
between $S_k(\G)\oplus\overline{S_k(\G)}$ and~$H^1_{\rm par}(\G,V)$ that is Hecke equivariant with respect to
the natural action of Hecke operators for~$\G$ on $H^1(\G)$.  But this is not the right way to get good arithmetic 
properties, because complex conjugation is not an algebraic operation.  To see this, think of the case~$k=2$; then
$S_2(\G)$ is isomorphic to the space of holomorphic 1-forms on the compact modular curve~$X_0(N)$, and the isomorphism 
just mentioned corresponds to the usual Hodge decomposition $H^1=H^{1,0}\oplus H^{0,1}$ 
valid for any compact Riemann surface, but this decomposition is transcendental and in algebraic contexts must 
be replaced by the isomorphism between $H^1$ and the quotient of the space of ``differentials of the second kind"
(meaning meromorphic 1-forms having residue~0 at all of their poles, or better, meromorphic 1-forms that are locally 
the derivatives of meromorphic functions) by the subspace of exact differentials (global derivatives of meromorphic 
functions).  For higher weights this means that we will have to work with the spaces $M_k^!(\G)$ and $ M_k^{\rm mer}(\G)$ 
of weakly holomorphic or meromorphic modular forms of weight~$k$ on~$\G$, rather than with just the holomorphic ones. 
(Recall that ``weakly holomorphic" means ``holomorphic in the upper half-plane but meromorphic at the cusps".)  
Instead of thinking of usual cusp forms as holomorphic modular forms that are small at infinity, we should see them 
as those that have vanishing constant terms at all cusps and hence can be integrated
any number of times.  We thus define the space $S_k^!(\G)$ of {\it weakly holomorphic cusp forms of weight~$k$} as 
the space of forms~in $M_k^!(\G)$ satisfying the same vanishing condition (i.e., having at every cusp a Fourier 
expansion of the form $\sum\limits_{n\gg-\infty}c_nq^n$ with $c_0=0$), and the larger space $S_k^{\rm mer}(\G)$ 
of {\it meromorphic cusp forms} or {\it cusp forms of the second kind} as the space of forms in $M_k^{\rm mer}(\G)$ 
that are locally $(k-1)$st derivatives (i.e., that satisfy the above condition at cusps and that have a Laurent series 
with vanishing coefficients of $(\tau-\tau_0)^{-i}$ for all $0<i<k$ at all poles~$\tau_0$ in the upper half-plane).  For $F$
belonging to either of these two spaces, we can define an Eichler integral~$\wt F$ of~$F$ just as before (but replacing 
``holomorphic" by ``meromorphic" in its definition in the case of $S_k^{\rm mer}(\G)$), and Bol's identity implies
just as before that $r_F$ is a cocycle (and in fact a parabolic cocycle) in~$H^1(\G,V)$.  We then get an
identification of the parabolic cohomology group $H_{\rm par}^1(\G,V)$ with the space 
$\Sm_k(\G) = S_k^{\rm mer}(\G)/D^{k-1}(M_{2-k}^{\rm mer}(\G))= S_k^!(\G)/D^{k-1}(M_{2-k}^!(\G))$. This space has
twice the dimension of $S_k(\G)$.
(More precisely, there is a short exact sequence of Hecke modules 
$0 \longrightarrow  S_k \longrightarrow  \mathbb{S}_k \longrightarrow  S_k^\vee \longrightarrow  0 $
given by mapping $(F,f) \in \mathbb{S}_k \times S_k$ to the sum over all points of $X_0(N)$
of the residues of $F(\tau) \tilde  f(\tau) d \tau$.)
This means that to each normalized Hecke eigenform $f$ in $S_k(\G)$ we can 
associate a meromorphic modular form $F$ (which can be chosen if we like to be weakly holomorphic, or 
even to be holomorphic at all cusps except~$\infty$) that has the same eigenvalues as~$f$ with respect to 
all Hecke operators modulo the $(k-1)$st derivatives of meromorphic (or weakly holomorphic) modular
forms of complementary weight~$2-k$.  However, since $\Sm_k$ has only a canonical filtration rather than a 
canonical splitting, this $F$ is only canonically defined up to a rational multiplicative factor (if we 
choose~$F$ to have rational Fourier coefficients) and the additition of a rational multiple of~$f$ and of a
$(k-1)$st derivative.   Once we have chosen~$F$, we fix the choice of its Eichler integral $\wt F$ by 
imposing $k-1$ linearly independent conditions with coefficients in~$\Q$ on the polynomials $r_F(\g)=\wt F|(1-\g)$
(e.g., by requiring that $r_F$ vanishes identically for $\g=T=\bigl(\smallmatrix 1&1\\0&1\endsmallmatrix\bigr)$
and that the constant term of the polynomial $r_F(\g)$ vanishes for some chosen $\g\not\in\langle T\rangle$; 
cf.~\cite{KlemmScheideggerZagier} for more details and examples). If we make these choices, then all values of the 
associated cocycle $r_F$ belong to $V(\Q)\eta_+\oplus V(\Q)i\eta_-$ for some real numbers $\eta_\pm$ that are 
well-defined up to multiplication by rational numbers and the addition of rational multiples of the original periods~$\om_\pm$.

We now make this explicit for the special case of the cusp form~$f_8$.  This is actually only one of 14
cases of cusp forms $f_N\in S_4(\G_0(N))$, corresponding to the 14 families of Calabi-Yau 3-folds whose 
corresponding Picard-Fuchs differential equations are hypergeometric (see the next section for more details), 
with values of $N$ ranging from 8 to~864, the most famous of these being the case $N=25$ corresponding to the 
hypergeometric function $\sum_n\frac{(5n)!}{n!^5}t^n$ and to the mirror quintic family that launched the whole 
field of mirror symmetry.  These families are discussed in detail in~\cite{KlemmScheideggerZagier}, where
an explicit choice of meromorphic cusp form $f_N^m$ with the same Hecke eigenvalues as $f_N$ is made for 
each of the 14 cases and the periods and quasiperiods computed numerically to high precision and shown to 
agree with entries of the transition matrices between bases of the spaces of solutions of the Picard-Fuchs
equations at the conifold point and at the ``MUM point" $t=0$.  For $N=8$ we can choose as our meromorphic
modular form $F$ with the same eigenvalues as~$\ff$ any of the three forms
$$ F_1(\tau) \,=\, \frac{\eta(2\tau)^{12}\eta(4\tau)^4}{\eta(8\tau)^8}\,,\quad
 F_2(\tau) \,=\,  \frac{\eta(2\tau)^{12}\eta(8\tau)^{16}}{\eta(4\tau)^{20}}\,,\quad
 F_3(\tau) \,=\, \frac{\eta(4\tau)^{28}}{\eta(2\tau)^4\eta(8\tau)^{16}}\,, $$
where $F_3$ differs from $F_1$ by $16\ff$ and from $256F_2$ by the third derivative of a form 
in $M_{-2}^!(\G_0(8))$.  The forms $F_2(\tau)$ and $F_3(\tau)$ are equal to $t_4(2\tau)\ff(\tau)$ and to
$t_4(2\tau)^{-1}\ff(\tau)$, respectively, with $t_4$ as in~\eqref{DEFt4}. It is now easy to give the proof of 
Theorem~3. For the proof of equation~\eqref{cB1andcA1} we used the second of equations~\eqref{cB1}, but here
the first is more convenient, since it tells us that we can compute the value of~$\cB(1)$ by an integral just like the
one used for the value~$\cB(0)=\cA(0)$ in equation~\eqref{BetaInteg}, but with the differential~$d\la/\la$ replaced 
by simply~$d\la$.  That means that we obtain~$\cB(1)$ simply by multiplying the integrand $(1-\tau)\ff(\tau/4)$
in~\eqref{cA0integral} by $\la(\tau)=\frac1{16}t_4(\tau/2)$.  This expresses $8\cB(1)$ as the period integral
$\int_0^2(1-\tau)F_2(\tau/4)\,d\tau$ of the weakly holomorphic cusp form~$F_2$ as desired, the integral being convergent
because $F_2(\tau)$ vanishes at the cusps $\tau=0$ and $\tau=1/2$ of $\G_0(8)$.

 \medskip
\section{Geometric interpretation: fibered motives}
We first describe in abstract terms the geometric/motivic ideas underlying the calculations
of the last sections, and then say in a little more detail how they look in our special case.

One way of proving an identity of the form 
$\int_{x=-\infty}^{\infty} \varphi  (x) dx = \int_{y=-\infty}^{\infty} \psi (y) dy$ 
is to show that there exists a space $\mathcal{S}$ such that $x$ and $y$ are two functions on it, 
and $\varphi (x)$ resp. $\psi (x)$ are the measures of their level subspaces; then by Fubini, both integrals 
are equal to the measure of $\mathcal{S}$.  This is a metaphoric rendering of a persistent theme 
in the arithmetician's study of differential equations that come from geometry as follows. 
Suppose we are given a hypergeometric variation of Hodge structures $\mathcal {V}$ on the torus 
$\Gm = \Spec \C [\lambda, \lambda^{-1}]$. 
That $\mathcal{V}$ is hypergeometric simply means that
the period(s) of $H = \mathcal{V}_{\lambda}$   can be written formally as
$\sum \mathbf{\Gamma}(k) \lambda^k$, where 
$\mathbf\Gamma(k) = \prod_i \Gamma (l_i(k))^{m_i}$
for some linear functions $l_i(k)=q_i k + r_i$ with $q_i \in \Z$, $r_i \in \Q$ and some 
exponents $m_i\in\Z$. The precise interpretation of the summation 
sign can be broader than the one we use in our examples where the products 
${\mathbf{\Gamma}} (k)$ will have the factor $k!$ 
in the denominator, as in the case of Gauss's $\, _{m+1}F_{m}$, 
so that the summation  can be chosen 
to be over $k \in \Z_{\ge 0}$. 

Now fix $\lambda = \lambda_0$. The Hodge structure 
$\mathcal{V}_{\lambda_0}$ plays the role of the space $S$ above, and we will 
see in a moment how two different lifts of the hypergeometric 
indices occurring in $\mathbf\Gamma(k)$ can be thought of 
as corresponding to the two functions $x,\, y$, providing 
a conceptual explanation of the identity 
$A_{-1/2}=B_{-1/2}$. 

 
Choose a \emph{lift} of $\mathbf{\Gamma}(k)$ to $\wt{\mathbf{\Gamma}}(n,k)$
given by the formula $\wt{\mathbf{\Gamma}}(n,k) = \prod_i \Gamma(\wt l_i(n,k))^{m_i}$,
where $\wt l_i(n,k)$ are now linear functions in \emph{two} variables 
of the form $\wt l_i(n,k)=p_i n + q_i k + r_i$, 
so that $\wt{\mathbf{\Gamma}}(0,k) = {\mathbf{\Gamma}}(k)$,
and set formally
\begin{equation} 
\label{Cn}
C_n \;=\;  \sum_{k} \wt{\mathbf\Gamma}(n,k) \lambda_0^k \quad  (n \in \C).
\end{equation}
(In our examples 
the lifts $\wt{\mathbf\Gamma}(n,k)$
have a $k!$ in the denominator and the sum 
over \mbox{$k \in \Z_{\ge 0}$} is convergent, but in general the summation 
in (\ref{Cn})  has to be understood formally.)
The expressions $C_n$ satisfy a finite-length recursion $R(C_n)=0$ for $n$ belonging to any coset 
$n_0+\Z$,
and there is a corresponding linear differential operator $L$ 
(the formal Mellin transform of~$R$) that formally annihilates the doubly infinite series
$\sum_{n\in n_0 \+ \Z} C_n t^n$ for any $n_0 \in\C$.
In the geometrically interesting situations when 
$\wt{\mathbf\Gamma}(n,k)$ is \emph{balanced}, i.e. when 
\mbox{$\sum_i m_i p_i = \sum_i m_i q_i = 0$,} 
this doubly infinite in $n$ series typically diverges for all values of $t$, 
but the truncated series $\Phi_{n_0}(t) = \sum_{n\in n_0 \+ \Z_{\ge0}} C_n t^n$ 
has finite radius of convergence and satisfies an inhomogeneous 
equation $L\Phi_{n_0}= r(t)$, where $r(t)$ is a finite Laurent series times 
$t^{n_0}$.

A formal analogue of Cauchy's formula would express $C_n$ as an integral
of the form $\int \Phi (t) t^{-n} \frac{dt}t$.
In particular, the same quantity 
\begin{equation} 
C_0 \;=\; \int \Phi (t)\, \frac{dt}{t},
\end{equation}
can be interpreted, depending on one's optic, as (a)~the value at~0 of the solution
of the recursion $R$
or, alternatively, (b)~as the period of the Hodge structure~$H$, or else 
(c)~a period in the Hodge structure arising in  the cohomology of the $t$-torus 
$\Spec \C\bigl[t,t^{-1}\bigr]$ with coefficients in the Hodge module $\mathcal H$ given by $L$. 
We will say that the Hodge structure $H$ is \emph{fibered out} by $t$ into a Hodge module 
$\mathcal H$ on $\Gm (t)$.  The mental picture behind this wording is that of an invertible 
function, or a ``unit'', on the motive $M$ that underlies~$H$ which turns it into a pencil of 
motives $\mathcal M$ over the $t$-torus. One of the benefits 
of the passage to a fibered motive from a variety equipped with a function fibering it into a 
pencil is that in real-life applications one will typically compactify and resolve in order to 
arrange a smooth proper morphism, in which case rubbish cohomology classes are often
acquired along the way that obscure the picture and cause a lot of struggle with unnecessary details. 

As we have already mentioned,
one can consider
 {\it two} different lifts of the same one-variable gamma product, corresponding 
to two different ``fibering-outs'' of the same 
motive over two tori $\Gm (t')$ and  $\Gm (t'')$. 
Recall that 
 the role of the space $\mathcal{S}$ 
is now played by the motive~$M$, while the units $t'$,\,$t''$ 
are analogues of $x$ and $y$. 
One should view the triple $(H,t',t'')$ as being
a (Hodge) correspondence between $H^1(\Gm (t'), \mathcal H')$ and $H^1(\Gm(t''),\mathcal H'')$,
and expect, generically, the orders of $R'$ and $R''$ to be equal to the rank of~$H$.  The values
at integer arguments of the solutions to $R'$ and $R''$ that are in the respective $\Q$-Betti spaces 
should span the same space of periods. This conceptual picture can be used as a tool to produce 
and prove statements about the equality of periods. Relatively straightforward 
in the cases of pure Hodge structures, such as period matrices of elliptic curves or periods 
of a conifold Calabi--Yau fiber of the Landau--Ginzburg model of a rank~1 Fano 4-fold 
(the case treated in this paper), this quickly becomes a source of much less trivial identities 
as one passes to \emph{iterated} variations of mixed Hodge structures and their special fibers. 

Summing up these observations, one anticipates a relation between the value of the {\it motivic gamma function}
\begin{equation}
\Gamma^{\text{mot}}_\mathcal{H} (s) \; := \; \int \Phi (t)\, t^s \, \frac{dt}{t}
\end{equation}
at $s=0$ and the entries of the period matrix of $H$ --- provided, 
of course, that one can make the meaning of the integral precise. 
We refer the reader to the recent paper \cite{BlochVlasenko2019} 
for the fundamentals of motivic gammas. 

A final remark here is that the comparison of higher \emph{derivatives} rather than values
of the motivic gamma functions at integer arguments arising from different fiberings $\mathcal{H}$ 
of the same Hodge structure $H$  
\begin{equation}
\frac{d^r \Gamma^{\text{mot}}_\mathcal{H} (s)}{ds^r} \;=\; \int \Phi (t)\, t^s \log (t)^r \, \frac{dt}{t}
\end{equation}
is an equally important subject, pertaining now to the study of mixed, rather than pure, motives.  
Special cases of Boyd's conjectures on Mahler measures of Laurent polynomials, for example, can be
interpreted as statements about the existence of a single mixed motive that can be fibered 
out in two specific ways as above. 
In what looks at first like a totally 
different setup, the \emph{Gamma Conjecture} relates the motivic gamma derivatives to the expansion 
coefficients of the ``gamma class" of a Fano variety whose regularized quantum differential equation 
is~$\mathcal H$.  In the final section of the paper we will give numerical evidence for this in the Ap\'ery case.  
      

 
The results of the preceding sections can be viewed 
as working out a case of the correspondence between 
two fiberings of a very specific $H$, and studying 
the respective motivic gammas. Explicitly, we start with  the gamma product 
$
 \mathbf{\Gamma}(k) = 256^{-k} \, \binom{2k}{k}^4  \;=\; \binom{-1/2}{k}^4  , \label{GammaProduct}
$
fix $\la_0 = 1$ and consider the two lifts given by 
\begin{equation}
 \mathbf{\wt{ \Gamma}}_A(n,k) \;=\; \binom{n-\frac{1}{2}}{k}^2 \binom{-n-\frac{1}{2}}{k}^2 \, 
\end{equation}
and
\begin{equation}
\mathbf{\wt \Gamma}_B(n,k) 
\;=\;
  4^{2n-1}\,{\binom {-\h}k}^2\biggl(\frac{\G(\h)}{\G(n-k+\frac{1}{2})\G(k-n+1)}\biggr)^2\,,
\end{equation}
corresponding to the two sequences ${A_{n-1/2}}$ and ${B_{n-1/2}}$  (cf. equations (\ref{AAx}) and~(\ref{BBy})).

The generating series of our gamma product 
\begin{equation} \label{Xi} 
\xi(\lambda) \;=\; \sum_{k=0}^\infty \mathbf{\Gamma}(k)\, \lambda^k \;=\;
\sum_{k=0}^\infty {\binom{2k}{k}}^4 \left( \frac{\la}{256}\right) ^k  
\end{equation} 
satisfies
the differential equation  
\begin{equation}
\label{HGdown} \bigl( D_\lambda^4 -  \lambda (D_\lambda+1/2)^4 \bigr) \, \xi (\lambda) \;=\; 0,  \quad \quad \bigl(D_\lambda = \lambda \frac{d}{d\lambda}\bigr), 
\end{equation}
one of the 14 hypergeometric DEs of order 4 that arise 
as regularized quantum differential equations of 
complete intersections in weighted projective spaces. 
The argument $\lambda_0 = 1$ is the \emph{conifold singularity}
of this equation. By Picard, hypergeometric differential equations 
with rational indices 
are known to be of geometric origin, or \emph{Picard--Fuchs}, 
which simply means that their solutions can be represented 
as integrals of certain algebraic differential forms 
over relative cycles in a pencil of varieties. 
In our case, the pullback of \eqref{HGdown}
under the map of tori given by $\lambda = 256 \, \Lambda^2$,
i.e. the differential equation 
\begin{equation}
\label{HGup} \bigl( D_\Lambda^4 - 256 \, \Lambda^2 (D_\Lambda+1)^4 \bigr) \, \Xi(\Lambda) \;=\; 0   \, ,
\end{equation}
corresponds to the pencil given by the Laurent polynomial on $\Gm^4$
\begin{equation} \Lambda^{-1} \;=\; \prod_{i=1}^4 \Bigl(x_i+ \frac{1}{x_i}\Bigr)\,.  \end{equation}  
This shows that the space of solutions to \eqref{HGdown} (or the dual space 
of flat sections) is a variation of $\Q$--Hodge structures.

The dimension of the space of analytic local solutions 
to \eqref{HGdown}  drops by $1$ at the conifold point. The 
corresponding period matrix is of rank $3$, but because of the symplectic polarization a rank~2 
$\Q$--Hodge structure $H$ splits off. The standard expectation is that it
comes from a modular \emph{newform}, so a correspondence should exist between the 
rigid Calabi--Yau threefold $C$ given by 
\begin{equation} \label{modu-KSZ} \prod_{i=0}^3 \Bigl(x_i+ \frac{1}{x_i}\Bigr) \;=\; 16\,. \end{equation}
and a Kuga-Sato variety. 
We refer the reader to papers 
\cite{CynkMeyer2007},
\cite{Meyer2005},
\cite{Rodriguez-Villegas2003},
\cite{SaitoYui2001},
\cite{Schoen1988},
\cite{Yui2003}
for various aspects of the link between rigid Calabi--Yau 
threefolds and modular forms,
and also~\cite{KlemmScheideggerZagier} for an explicit construction 
of the correspondence with a Kuga-Sato variety in the case~(\ref{modu-KSZ}).

The rank~2 motive $M$ that underlies $H$  can be fibered out 
``in different directions'' that correspond to the choice of the lifts. 
In our situation one can be looking for special choices that turn $\mathcal{H}$ 
into a \emph{modular} variation of Hodge structures. 
This means that in 
the differential equation $\mathcal{L} \Phi_{1/2} (t) = 0$ that 
controls $\mathcal{H}$, the solution $\Phi_{1/2} (t)$ can be chosen to be a 
weight~2 modular form, and $t$ a Hauptmodul
for the same congruence
subgroup.
The modular parameter $\tau$ 
could then be interpreted as the ratio $\frac{\Phi_{1/2}^*(t)}{\Phi_{1/2}(t)}$ of the 
normalized log to the analytic solution around $t=0$; 
the functions ${\Phi_{1/2}(t(\tau))}, \, \tau {\Phi_{1/2}(t(\tau))} = {\Phi_{1/2}^*(t(\tau))}$
and $\tau^2 {\Phi_{1/2}(t(\tau))}$ 
form a basis of the space of local solutions of this differential equation
around a cusp.  
Among the entries of the period matrix of $H$ should then be the value 
of the \emph{Eichler integral} of the weight~4 form $\Phi(t(\tau))t'(\tau)$. This can be 
made even more precise if one notices that the essential ingredient of the $L$-function
of $C$ computed by the point count is the Mellin transform of the level~4 form~$\ff(\tau)$.  

In our examples, the weight~2 modular forms
appearing  in \eqref{ApMod} and \eqref{BMod}
arise exactly in this way 
from the two lifts of $\mathbf{\Gamma}(k)$
indicated above, as summed up in the following table: 
\bigskip 
\begin{center}
\begin{tabular}{ c|c|c|c} 
& $ \mathbf{\wt{ \Gamma}}(n,k)$ & $t(\tau)$ & $\Phi_{1/2}(t(\tau))$  \\ 
 \hline
$ \rule[-2em]{0pt}{5em}\text{Case } A \phantom{mm}$  &  $
  \binom{n-1/2}{k}^2 \binom{-n-1/2}{k}^2 \, 
$ & $\dfrac{\eta(\tau)^{12}\,\eta(6\tau)^{12}}{\eta(2\tau)^{12}\,\eta(3\tau)^{12}}$ & $ \dfrac{\eta(2\tau)^7\eta(3\tau)^7}{\eta(\tau)^5\,\eta(6\tau)^5}$\\ 
\hline
$\rule[-2em]{0pt}{5em}  \text{Case } B \phantom{mm}$  &  $
2^{4n-2} \binom{-1/2}{k}^2 \binom{-1/2}{k-n}^2  
$ & $\dfrac{\eta(\tau)^8\eta(4\tau)^{16}}{\eta(2\tau)^{24}} $ & $\dfrac{\eta(2\tau)^{20}}{\eta(\tau)^8\eta(4\tau)^8}$

\end{tabular}

\end{center}
\medskip
and are  
related to the weight~4 form~$\ff$ by
\begin{equation} f_6(\tau)\,t_6(\tau)'/t_6(\tau)^{1/2} \;=\; 
\ff(\tau) \m 9\,\ff(3\tau),\qquad f_4(\tau)\,t_4(\tau)'/t_4(\tau)^{1/2}\;=\; \ff(\tau).\end{equation}
In contrast,
the forms $f_N(\tau)t_N'(\tau)$ and $f_N(\tau)t_N'(\tau)/t_N(\tau)$ for $N=4$ or~6 are weight~4
Eisenstein series whose periods involve the number~$\z(3)$ occurring in the Gamma Conjecture \cite{GolyshevZagier2016}.
In our language, these would correspond to ``fibering--outs'' of 
a Hodge--Tate structure that arises in a special fiber in a highly 
reducible hypergeometric variation of mixed Hodge--Tate 
structures. Even though these two rank 2 Hodge structures
appear to be completely dissimilar at first sight, 
both are members in a continuous family of integrals varying 
with the parameter $s$ of the motivic gamma function. 
The reason to pay special attention to the cases where $s$ is an integer or half--integer
is that we want our Hodge structures to be defined over $\Q$, which requires 
$\exp (2 \pi i s) \in \Q$.  
    
 \section{Asymptotics} 
In this section we discuss the asymptotic properties of various solutions of the recursions~\eqref{AJ} and~\eqref{AE}. 
The two cases are very different.  In the Ap\'ery case, if we consider the homogeneous version of the shifted recursion~\eqref{ShiftedRec} 
(with $\cA(x)$ replaced by an unknown solution $F(x)$ and the right-hand side replaced by~0) and assume as Ansatz that 
$F(x)\sim C^xx^\mu P(1/x)$ for $x$ large, where $C$ and $\mu$ are constants and $P$ is a power series, then we find two asymptotic
solutions, an exponentially large one and an exponentially small one, given by
\begin{equation}\label{LargeSmall} \LL_A(x) \;\sim\; \frac{C_0^x}{x^{3/2}}\,P\Bigl(\frac1{64x\sqrt2}\Bigr)\,,\qquad 
 \SSS_A(x) \;\sim\; \frac{C_0^{-x}}{x^{3/2}}\,P\Bigl(-\frac1{64x\sqrt2}\Bigr)\,, \end{equation}
where $C_0=17+12\sqrt2=33.9705\cdots$ and $P(X)\in\Q[[X]]$ is a power series beginning
$$ P(X) \;=\; 1 \+ 30X \+ 274X^2 \m 17132X^3 \m 444234X^4 \+ 41390724X^5 \+ \cdots \,, $$
In particular, since the solution space of the recursion in a fixed residue class $\Z+x_0$ ($x_0\in\C$)
is 2-dimensional, we deduce that any unbounded solution of the recursion in this residue class grows exponentially and 
is asymptotically equal to a multiple of $\LL_A(x)$ to all orders in~$1/x$, while any bounded solution decays exponentially 
and is asymptotically equal to a multiple of $\SSS_A(x)$ to all orders.  For example, from Stirling's formula and the Euler-Maclaurin formula
one finds easily that $\cA(x)\sim2^{-9/4}\pi^{-3/2}\LL_A(x)$ to all orders as~$x\to\infty$.  Notice that the small solution
is asymptotically about~$1154^n$ times smaller than the large one. It is this huge dichotomy that permitted Ap\'ery to prove the 
irrationality of~$\z(3)$, since it implies that the ratio of the two solutions of~\eqref{AJ} with initial values $(0,1)$
and~$(1,5)$ tends to its limit $\frac16\,\z(3)$ with great exponential rapidity.

For the $B$-case the situation is different because if we make the same Ansatz $C^xx^\mu P(1/x)$ for the recursion~\eqref{AE},
then the corresponding characteristic equation is \hbox{$C^2-2C+1$}, which has a double root at $C=1$ (or $C=16$ if we consider~\eqref{AK} instead),
as opposed to the two different roots $C_0^{\pm1}$ of the corresponding equation \mbox{$C^2-34C+1=0$} in the $A$-case.  This means
that there are now two asymptotic solutions of comparable size: a ``small" solution
\begin{equation}\label{AF} \SSS_B(x) \;\sim\; \frac1{x} \m \frac1{2^4\,x^3} \+ \frac{17}{2^{10}\,x^5} 
     \m \frac{169}{2^{14}\,x^7} \+\frac{50777}{2^{22}\,x^9} \m \cdots \end{equation}
in $\frac1x\Q[[\frac1{x^2}]]$  and a ``large" (or ``logarithmic") solution
\begin{equation}\label{AG} \LL_B(x) \;\sim\; \bigl(\log(16x)+\g\bigr)\,\SSS_B(x)\+ \frac{5}{24\,x^3} 
      \m\frac{1219}{30720\,x^5} \+ \frac{304469}{10321920\,x^7}\m\cdots \end{equation}
in $(\log x + \kappa)\,\SSS_B(x)\+\frac1{x^3}\Q[[\frac1{x^2}]]$, where the constant $\kappa=\log16+\g$ ($\g$~= Euler's constant) 
has been chosen for later convenience.  This in turn means that, whereas in the $A$-case we get only a {\it filtration} of the 
2-dimensional solution space, with a uniquely defined solution that is asymptotically equal to~$\SSS_A(x)$ but whose second solution 
 that is asymptotically equal to~ $\sim\LL_A(x)$ that is defined only up to multiples of the first, in the $B$-case we get a 
well-defined {\it basis} of two solutions $S(x)$ and $L(x)$ having the asymptotics $\SSS_B(x)$ and $\LL_B(x)$, respectively. 
More precisely, this means that the recursion relation~\eqref{AE} has two meromorphic solutions~$L_B(x)$ and $S_B(x)$ (from now on
we omit the subscript~``$B$" for typographical convenience) determined uniquely by the requirements that
$$  S(x) \;=\;  \frac{1\+\text{\rm o}(1)}x\,, \qquad L(x) \;=\; \frac{\log(16x)\+\g\+\text{\rm o}(1)}x $$
as $x\to\infty$ with $x$ real.  These functions then satisfy the asymptotic formulas $S(x)\sim \SSS_B(x)$ and $L(x)\sim\LL_B(x)$ 
to all orders in~$1/x$ as $x\to\infty$ with $x$ real or as $\Re(x)\to\infty$ with $\Im(x)$ fixed, where $\SSS_B(x)$ and $\LL_B(x)$ are
the asymptotic solutions given above, and have poles (of order at most~3) at $x=-\h,\,-\frac32,\dots$ as their only singularities.

It follows that {\it any} solution $F(x)$ of the recursion on a fixed class in $\C/\Z$ is a linear combination $\lambda L(x)+\mu S(x)$, 
where the coefficients $\lambda$ and $\mu$ are determined by $xF(x)=\lambda(\log(16x)+\g)+\mu+\ooo(1)$ as~$x\to\infty$.  In particular, it turns out
that both $16^{-n}B_n$ and $\cB(n)$ for $n\in\Z_{\ge0}$ are multiples of~$L(x)$ (where $x=n+\h$ or~$n$, respectively), with the coefficient 
of~$S(x)$ being~0 in both cases (this was the reason for the choice of the constant $\log(16)+\g$ in the definition of~$L(x)$), but 
that $\cB(x)$ for non-half-integral values of~$x$ is {\it not} a multiple of~$L(x)$. The full result is given as follows.

\begin{theorem} For general complex values of~$x$, the solutions $\cB(x)$ and $\cB(-x)$ are given in terms of $L(x)$ and $S(x)$ by
\begin{equation}\label{cBasymp} \cB(x) \;=\; \frac{3-\cos 2\pi x}{2\pi^2}\,L(x) \+ \frac{\sin 2\pi x}{2\pi}\,S(x)\,, \qquad \cB(-x) \;=\; \frac{\cos^2\pi x}{\pi^2}\,L(x)\,.\end{equation}
In particular, the numbers $16^{-n}B_n$ and $\cB(n)$ $(n\in\Z_{\ge0})$ are multiples of the ``large" solution:
\begin{equation}\label{Large}  B_n \;=\; \frac2{\pi^2}L(n+\h)\,, \qquad \cB(n)\;=\;\frac1{\pi^2}L(n) \qquad(n=0,\,1,\,2,\dots)\;. \end{equation}
\end{theorem}
\begin{corollary}
The functions $L(-x)$ and $S(-x)$ are given in terms of $L(x)$ and $S(x)$ by
$$ \begin{pmatrix} L(-x)/2\pi^2 \\ S(-x)/2\pi \end{pmatrix}  \;=\; \frac1{2\cos^2\pi x}\,
\begin{pmatrix} 3-\cos 2\pi x  & \sin 2\pi x \\ 8\,\tan \pi x & 3-\cos 2\pi x \end{pmatrix}
\begin{pmatrix} L(x)/2\pi^2 \\ S(x)/2\pi \end{pmatrix} \;. $$
\end{corollary}

\noindent{\it Proof.} We only need to prove equation~\eqref{cBasymp}, since~\eqref{Large} is just its specialization to half-integral~$x$.
Since both $\cB(x)$ and $\cB(-x)$ are solutions of the equation, by the previous remarks it suffices to prove the asymptotic formulas
\begin{equation}\label{AH} \aligned x\,\cB(x)& \;=\;\frac{3+\cos2\pi x}{2\pi^2}\,\bigl(\log(16x)+\g\bigr)\,\log x \+ \frac{\sin2\pi x}{2\pi}\+\ooo(1)\,, \\
x\,\cB(-x) &\;=\; \frac{\cos^2\pi x}{\pi^2}\,\bigl(\log(16x)+\g\bigr)\,\log x \+ \ooo(1) \endaligned \end{equation}
as $x\to\infty$.  We prove the second statement only, the proof of the first being similar. Suppose that $|x|\to\infty$ with a fixed 
argument between $-\pi$ and~$\pi$.  From the formula $\G(x)\G(1-x)=\frac\pi{\sin\pi x}$ we have
$$ \frac\pi{\cos^2\pi x} \,\cB(-x) \;=\; \sum_{k=0}^\infty {\binom {-\h}k}^2\biggl(\frac{\G(x+k+\h)}{\G(k+1-x)}\biggr)^2
\;=\; \sum_{k=0}^\infty {\binom {-\h}k}^2\biggl(\frac1{x+k}\+ \OOO\Bigl(\frac1{(x+k)^2}\Bigr)\biggr) \,.$$
The second term can be estimated easily (by using that the binomial coefficient as $\OOO(k^{-1/2})$ and
breaking up the sum into $k\le|x|$ and $k>|x|$) as $\OOO(\frac{\log x}{x^2})$, and the first is equal to
$$\sum_{k=0}^\infty\biggl( {\binom {-\h}k}^2 \m \frac1{\pi(k+1)}\biggr)\,\frac1{x+k} 
 \+\frac1{\pi(x-1)}\sum_{k=0}^\infty\biggl(\frac1{k+1}\m\frac1{k+x}\biggr)
  \;=\;  \frac{c+\log x+\g+\ooo(1)}{\pi x}$$
for a certain constant $c$. To calculate it, we use the modular parametrization: 
\begin{align*} 
c &\;=\; \sum_{k=0}^\infty\Bigl( \pi\,{\binom {-\h}k}^2\m \frac1{k+1}\Bigr)
  \;=\; \lim_{\lambda\to1}\Bigl(\pi\,\bb(\lambda) \m \frac1\lambda\,\log\Bigl(\frac1{1-\lambda}\Bigr)\Bigr) \\
 & \;=\; \lim_{\tau\to0}\Bigl(\pi\,\theta_3(\tau)^2 \m \frac1{\lambda(\tau)}\,\log\Bigl(\frac1{1-\lambda(\tau)}\Bigr)\Bigr) \\
 & \;=\; \lim_{\tau\to\infty}\Bigl(-\pi i\tau\,\theta_3(\tau)^2 \+ \frac{1}{1\m\lambda(\tau)}\,\log \lambda(\tau)\biggr) \;=\; \log 16\,,  
\end{align*} 
where we have used the modularity properties $\vth_3\bigl(-\frac 1\tau)^2 = -i\tau\vth_3(\tau)^2$ and $\lambda\bigl(-\frac1\tau\bigr)=1-\lambda(\tau)$.
Finally, the corollary follows directly from the theorem, since
$$  \begin{pmatrix} 3-\cos 2\pi x& -\sin 2\pi x \\ 2\,\cos^2\pi x & 0 \end{pmatrix} ^{-1} \!
   \begin{pmatrix}  2\,\cos^2\pi x & 0  \\ 3-\cos 2\pi x& \sin 2\pi x \end{pmatrix} 
  \;=\; \frac1{2\,\cos^2\pi x}\,\begin{pmatrix} 3-\cos 2\pi x  & \sin 2\pi x \\ 8\,\tan \pi x & 3-\cos 2\pi x \end{pmatrix} \;. $$

\section{The kappa series and the Gamma Conjecture}
In \cite{GolyshevZagier2016}, we considered the Frobenius deformation of  Ap\'ery's recursion \eqref{AJ}, i.e.,
$$
 (n+\varepsilon+1)^3\,A_{n+1}(\varepsilon) \m P(n+\varepsilon)A_n(\varepsilon) \+ (n+\varepsilon)^3\,A_{n-1}(\varepsilon) \;=\;0\;  
$$
with $P(x)=34x^3+51x^2+27x+5$, and its solution, the sequence of 
functions $ A_n(\varepsilon)$
defined by the initial condition $A_{-1}(\varepsilon)=0$, $A_0(\varepsilon)=1$.
We then defined the kappa function by
$$
\kappa^0(\varepsilon) \;=\; \lim\limits_{n\to\infty} \frac{A_n(\varepsilon)}{2^{-9/4}\pi^{-3/2} \mathcal{L}_A(n+1/2+\varepsilon)}.
$$
The kappa function is meromorphic
in the complex plane and holomorphic at $0$.
Expand kappa in Taylor series around $0$
as $\sum_{j=0}^\infty \kappa^{(0)}_j \varepsilon^j$.
The question was raised in \cite{GolyshevZagier2016} and later 
investigated in~\cite{BlochVlasenko2019} of whether
the expansion coefficients $\kappa^{(0)}_j$ are in the algebra 
of multiple zeta (or merely Riemann zeta) values.
We found numerically with high precision that
\begin{multline} \label{K00}
\kappa^{(0)}_0 \;=\;1,\, 
 \quad \kappa^{(0)}_1 \;=\;0,\,
 \quad \kappa^{(0)}_2 \;=\;-\frac{1}{3}\,{\pi }^{2},\,
 \quad \kappa^{(0)}_3 \;=\;{\frac {17}{6}}\,\zeta  \left( 3 \right),\, \\
 \quad \kappa^{(0)}_4 \;=\;\frac{1}{45} \,{\pi }^{4},\, 
 \quad \kappa^{(0)}_5 \;=\;-{\frac {17}{18}}\,{\pi }^{2
}\zeta  \left( 3 \right) +\frac{7}{3}\,\zeta  \left( 5 \right),\, \\
\quad  \kappa^{(0)}_6={\frac {4}{945}}\,{\pi }^{6}+4\, \zeta
 \left( 3 \right)  ^{2},\,
  \quad \kappa^{(0)}_7 \;=\;-{\frac {7}{9}}\,{\pi }^{2} \zeta  \left( 5 \right) +
{\frac {7}{108}}\,{\pi }^{4}\zeta  \left( 3 \right) - \frac{5}{3} \,\zeta
 \left( 7 \right),\,
\\ \kappa^{(0)}_8 \;=\;-{\frac {11}{37800}}\,{\pi }^{8}+6\,\zeta  \left( 5
 \right) \zeta  \left( 3 \right) - \frac{4}{3}  \,  {\pi }^{2} \zeta  \left( 3
 \right)  ^{2},\,
\\  \kappa^{(0)}_9 \;=\;{\frac {8}{9}}\,\zeta  \left( 9 \right) +{\frac {34}
{9}}\,  \zeta  \left( 3 \right)   ^{3}+  \frac{5}{9}   \,{\pi }^{2}\zeta
 \left( 7 \right) +{\frac {149}{11340}}\,{\pi }^{6}\zeta
 \left( 3 \right) +{\frac {5}{54}}\, {\pi }^{4} \zeta  \left( 5 \right) ,\, 
\\  \kappa^{(0)}_{10} \;=\;-{\frac {107}{249480}}\,{\pi }^{10}-4\, 
\zeta  \left( 5 \right)   ^{2}-8\,\zeta  \left( 3 \right) \zeta  \left( 7 \right)
 +{\frac {4}{45}}\,{\pi }^{4}  \zeta
 \left( 3 \right)   ^{2}-2\,{\pi }^{2}\zeta
 \left( 3 \right)  \zeta  \left( 5 \right) , 
\end{multline}
involving only Riemann zeta values,
while $\ka_{11}^{(0)}$ is given numerically by
\begin{multline*} 
\kappa^{(0)}_{11} \;=\;
-{\frac {503}{680400}}\,{\pi }^{8}\zeta  \left( 3 \right) +{\frac {199
}{5670}}\,{\pi }^{6}\zeta  \left( 5 \right) +{\frac {49}{270}}\,{\pi }
^{4}\zeta  \left( 7 \right) 
-{\frac {34}{27}}\,{\pi }^{2} 
\zeta  \left( 3 \right)   ^{3} \\
-{\frac {8}{27}}\,{\pi }^{2}
\zeta  \left( 9 \right) +{\frac {28}{3}}\, \zeta  \left( 3
 \right)  ^{2}\zeta  \left( 5 \right) +66\,\zeta  \left( 11
 \right) + \frac{2}{3} \, \zeta  \left( 3,5,3 \right). \phantom{XXXXXXX}
\end{multline*}

The function defined by the expression $\kappa^0(- \eps)/\eps^3$ can be shown \cite{BlochVlasenko2019} to satisfy the original homogeneous 
(i.e. without the correction term)
three--term recursion \eqref{AJ}. It follows that the  derivatives of $\kappa^0$ at any integer
argument $n$ can be expressed linearly in terms of the two sets of derivatives,
at $0$ \emph{and} at~$1$. 
Computing many more derivatives at $0$ numerically and 
using them to compute near $\eps = 1$
(for this we use the fact that the function 
$\kappa^0(\eps)/\Gamma(1+\eps)^3$ is entire and therefore has an
extremely rapidly converging Taylor expansion), 
we find that for the first five expansion coefficients
$$
\kappa^{(0)}_{1,j} := \frac{1}{j!} \frac{d^j}{dx^j} \biggl(\frac{\kappa^{(0)}(x)}{(1+x)^{3}} \biggr) \, \biggl|_{x=1}
$$
one has
\begin{align} \label{K01}
\kappa^{(0)}_{1,0} \;=\;\frac{1}{6}  \,\zeta  \left( 3 \right)  ,
 \quad \kappa^{(0)}_{1,1} \;=\;-{\frac {1}{90}}\,{\pi }^{4},\,
 \quad \kappa^{(0)}_{1,2} \;=\;- \frac{1}{18} \,{\pi }^{2} \zeta  \left( 3 \right) +\frac{11}{3}\,
\zeta  \left( 5 \right),\,
\\
\kappa^{(0)}_{1,3} \;=\;-{\frac {13}{1890}}\,{\pi }^{6},\,
 \quad \kappa^{(0)}_{1,4} \;=\;{\frac {59}{3}}\,\zeta  \left( 7 \right) +{
\frac {19}{540}}\,{\pi }^{4}\zeta  \left( 3 \right) -{\frac {11}{9}}\,
{\pi }^{2} \zeta  \left( 5 \right) ,
\end{align}
but
$$
 \kappa^{(0)}_{1,5} \;=\;-{\frac {29}{56700}}\,{\pi }^{8}-10\,\zeta
 \left( 3 \right) \zeta  \left( 5 \right)  -4\zeta \left( 3,5 \right).
$$
We have introduced the factor
$(1+x)^{-3}$ 
because it turns out to make all of the conjectural 
formulas above homogeneous in the weight. In  fact, the first identity in 
(\ref{K01}) is a rewording of Ap\'ery's original discovery, see  \cite{BlochVlasenko2019}. 
Reasoning in the spirit of the Gamma Conjecture,
we predicted in \cite{GolyshevZagier2016} that
the top--weight components of the leading
expansion coefficients of kappa at $0$ and at $1$,
namely the numbers $\ka^{(0)}_{j}$ for $ j = 0, \dots , 10$
and $\kappa^{(0)}_{1,j}$ for $j= 0, \dots, 4$
should be equal to the expansion coefficients
of the normalized gamma class of the orthogonal
Grassmannian $OG(5,10)$ of isotropic $5$-planes in a
$10$-dimensional space endowed with a non-degenerate
orthogonal form.

Recall that the normalized gamma class of a smooth variety $V$ of 
index $d$ is given by the formula
$$\widehat{\Gamma}^{\text{norm}} (V) \;=\; \dfrac{\prod_{\alpha} \Gamma(1+r_\alpha)}{\Gamma( 1+(-K_V)/d)^d} ,$$
where the product in the numerator is taken with respect to the formal Chern roots $r_\alpha$ of the tangent bundle $\mathcal{T}_V$
(so that $c(T_V)=\prod_\alpha(1+r_\alpha)$).
We will explain in the particular case of $OG(5,10)$ how to compute the expansion coefficients of the normalized 
gamma class with respect to the elements of the Lefschetz basis.  
We warn the reader that the normalized gamma class should 
not be confused with the regularized gamma class considered 
by the authors in \cite{GolyshevZagier2016} and given by the formula 
$\widehat{\Gamma}^{\text{reg}} (V) \;=\; \bigl({\prod_{\alpha} \Gamma(1+r_\alpha)}\bigr)/{\Gamma( 1+(-K_V))}$. In the context of
Fano topology, both relate naturally to the gamma class of Calabi--Yau subvarieties: in the former case, of a $d$--fold complete intersection, and in the latter,  of a Calabi--Yau anticanonical section hypersurface. It is the normalized gamma class that, by definition, is essentialy stable under passage to hyperplane section.  

Similarly to the case of the ordinary Grassmannian, where the classes of the Schubert cells 
are numbered by partitions and expressed as Schur polynomials in the 
Chern roots of the universal quotient bundle $Q$, the cell classes of $OG$ 
are numbered by \emph{strict} partitions---in our case, those that lie within the 4-by-4 box. As in 
the ordinary case, there is a map from the ring of  symmetric
functions to the cohomology ring, but the formulas are more complicated. 
In particular, the elementary symmetric functions $e_i,  \, i = 1, \dots, 4$ evaluated on the 
Chern roots of $Q$ give twice the cell classes that correspond 
to the special (one-strip) partitions: $\tau_{i} = \frac{c_i(Q)}{2}$, while 
the class $c_5(Q)$ vanishes. Thus, the $\tau_i$'s generate the cohomology as algebra, the ideal of relations           
being generated by the \emph{Pragacz--Ratajski polynomials}:  
$$ \label{clgiam1}
\wt{P}_{i,i}(X) \;=\;  \tau_i^2 \+ 2\sum_{k=1}^{i-1}(-1)^k \tau_{i+k} \tau_{i-k} \+(-1)^i  \tau_{2i} 
 \qquad (i=1, \dots, 4). $$
Looking (e.g. with the help of Gr\"obner bases) for a linear combination of $\tau_1^3$ and $\tau_3$ 
annihilated by multiplication by the class 
$\tau_1^5$, we find that the class  $\Prim = 7 \tau_1^3 -12 \tau_3$ is primitive in weight~3,  in the sense that 
$\Prim \tau_1^5=0, \, \Prim \tau_1^4 \ne 0$. 
Therefore, the two Lefschetz $sl_2$--submodules in $H^*(OG(5,10),\Q)$ are 
spanned by $\{ \tau_1^i \mid i=0,\dots , 10 \}$ and   
$\{ \Prim \tau_1^i \mid i=0, \dots , 4 \}$ respectively. 
Passing from the full Chern class 
$\mathrm{Ch}(Q,t) \;=\; 1+ \sum_{i=1}^5 {2\tau_i}t^i$ 
to the Chern character in the usual way, using the isomorphism $\mathcal{T}_{OG} = \bigwedge ^2 Q$, 
which implies the relation between the Chern characters 
$$\Ch(\mathcal{T}_{OG},t) \;=\; \frac{1}{2} \Bigl(\Ch(Q,t)^2 - \Ch(Q,2t) \Bigr),$$ 
and expanding 
$$\Ch(\mathcal{T}_{OG},t) \;=\; \sum_{i \ge 0} \frac{p_i}{i!} t^i,$$
we compute the normalized gamma class with the formula
$$
\widehat{\Gamma}^{\text{norm}}(OG(5,10)) \;=\; \dfrac{ \exp \Bigl( -p_1 \gamma  t 
+ \sum_{i=2}^{10} (-1)^i \,\frac{\zeta(i)}{i}  \, p_i  t^i \Bigr) }
{\exp  \Bigl( -\gamma \tau_1 t 
+ \sum_{i=2}^{10} (-1)^i \, \frac{\zeta(i)}{i} \tau_1^i \frac{t^i}{i} \Bigr)^8 }  \quad (\text{mod } t^{11}) \, .
$$
By straightforward computation we prove:
\begin{theorem}
The expansion coefficients
of the normalized gamma class of $OG(5,10)$ 
with respect to the Lefschetz basis
$$\{ \tau_1^i \mid i=0,\dots , 10 \} \cup 
\{ \Prim \tau_1^i \mid i=0, \dots , 4 \}$$
coincide with 
the top--weight components of the leading
expansion coefficients of the kappa function at~$0$ and at~$1$
as found experimentally and given above in  \eqref{K00},\! \eqref{K01}:
$$
\widehat{\Gamma}^{\text{norm}}(OG(5,10))\, \;=\; \, 
1 \+ \sum_{i=2}^{10} \kappa^{(0)}_i  \tau_1^i t^i
\+ \sum_{j=0}^4 \kappa^{(0)}_{1,j} \Prim \tau_1^{j} t^{j+3}  
. 
$$
\end{theorem} 

 
 \section*{Afterword}
We would like to thank Spencer Bloch, Masha Vlasenko, and Duco van Straten
for many illuminating discussions in connection with the present paper.  All three
are members of the group that undertook to pursue the subject of motivic gamma functions.
We would like to mention in particular two papers that followed our discussions at meetings in 
Bures and at Herstmonceux Castle in June 2016. In \cite{BlochVlasenko2019}, Bloch and Vlasenko work 
out the relation between the Mellin transform associated to a solution of a $DN$-type equation
with the corresponding Ap\'ery series, and link the Ap\'ery constants with periods of limiting mixed Hodge
structures. In \cite{GolyshevvanStratenZagier2019}, the present authors and van Straten consider dimensional 
interpolation of the Riemann-Roch--Hirzebruch formalism for Grassmannians and indicate how Ap\'ery-like 
numbers can be interpolated with respect to the dimensional parameter as well as the index.  In this context 
we refer the reader to \cite{LoeserSabbah1991a} for the foundations of $D$-modules on tori and the algebraic
Mellin transform. We would also like to thank Andrew Kresch for communicating to us facts about cohomology 
of the orthogonal Grassmannian and for organizing the activity in Konolfingen
in Switzerland where parts of this work were first presented.

Above all, we want to express our gratitude to Boris Dubrovin, to whose memory this paper is
dedicated, for the inspiration that he provided for us and for many other members of our generation,
as both a mathematician and a person of gigantic stature.  It was an honor and a pleasure to learn 
from him and to work with him, and his premature passing leaves a gap in our lives.  In view of his many 
contributions to the theory of mirror symmetry, and in particular his proof of the Gamma Conjecture for
projective spaces of every dimension, we hope that this paper is an appropriate tribute to him.

 
\nocite{Beukers1987,BeukersPeters1984}
\nocite{Golyshev2007}

\bibliographystyle{plainurl}
\bibliography{references}

% \begin{thebibliography}{10}

% \bibitem{Beukers1987}
% {\sc Beukers, F.}
% \newblock {Irrationality proofs using modular forms.}
% \newblock {Journ\'ees arithm\'etiques, Besan\c{c}on/France 1985, Ast\'erisque
%   147/148 (1987), 271-283.}

% \bibitem{BeukersPeters1984}
% {\sc Beukers, F., and Peters, C.}
% \newblock {A family of K3 surfaces and $\zeta(3)$.}
% \newblock {\em J. Reine Angew. Math. 351\/} (1984), 42--54.

% \bibitem{BlochGolyshev}
% {\sc Bloch, S., and Golyshev, V.}
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% \end{thebibliography}

\end{document}
