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An Ansatz for the asymptotics of hypergeometric multisums. (English) Zbl 1170.34059

An (\(r\)-)balanced hypergeometric term \(t=t(n,\mathbf k)\), where \(n\in\mathbb Z_{\geq0}\) and \(\mathbf k=(k_1,\dots,k_r)\in\mathbb Z^r\), is an expression of the form \[ t(n,\mathbf k) =C_0^n\prod_{l=1}^rC_l^{r_l}\prod_{j=1}^JA_j(n,\mathbf k)!^{\epsilon_j}, \] with \(C_l\) algebraic for \(l=0,1,\dots,r\), \(\epsilon_j\in\{\pm1\}\) and \(A_j\) homogeneous linear forms with coefficients from \(\mathbb Z\) in \(n,k_1,\dots,k_r\) for \(j=1,\dots,J\) subject to the balance condition \[ \sum_{j=1}^J\epsilon_jA_j(n,\mathbf k)=0. \] For a balanced term \(t\), define its Newton polytope \[ P_t=\{\mathbf x\in\mathbb R^r:A_j(1,x_1,\dots,x_r)\geq0 \;\text{for}\; j=1,\dots,J\}\in\mathbb R^r \] and assume that \(P_t\) is a compact rational convex polytope in \(\mathbb R^r\) with non-empty interior. Then \(nP_t\cap\mathbb Z^r\) expresses the support of \(t(n,\mathbf k)\) and the sequence \[ a_n(t)=\sum_{\mathbf k\in nP_t\cap\mathbb Z^r}t(n,\mathbf k) \] is well defined, since the sum is finite for every \(n\in\mathbb Z_{\geq0}\). Under these assumptions, the generating series \[ G_t(z)=\sum_{n=0}^\infty a_n(t)z^n\in\overline{\mathbb Q}[[z]], \] where \(\overline{\mathbb Q}\) denotes the field of algebraic numbers, is a \(G\)-function as the author shows in his preprint [arXiv:0708.4354[math.CO] (2007/08)]. In particular, this means that \(G_t(z)\) satisfies a Picard–Fuchs differential equationan equation whose singular points are all regular. Let \(\Sigma_t\) denote the set of the singular points (without the point at infinity) of this differential equation.
A practical (but not usual from the reviewer’s point of view) way for computing the asymptotic expansion for sequences \(a_n(t)\) is to find a linear recurrence (using, for example, the Gosper–Zeilberger algorithm of creative telescoping), convert it into a linear differential equation with polynomial coefficients for \(G_t(z)\) and determine the set of roots of the coefficient of the leading derivative in this equation; then one of the roots characterizes behavior of \(|a_n(t)|^{1/n}\) (and, in many cases, of \(a_{n+1}(t)/a_n(t)\)) as \(n\to\infty\).
Since the set of roots forms the set \(\Sigma_t\) of singular points, it looks desirable to compute it without actual computing the differential equation for \(G_t(z)\). The main result of the article under review is a construction of a finite set \(S_t\subset\overline{\mathbb Q}\) which conjecturally include \(\Sigma_t\). The conjecture is supported by showing that \(\Sigma_t\subset S_t\) for two classes of balanced terms \(t\).
In his attempt to build up a philosophy on determining the set of singular points for the Picard–Fuchs differential equation satisfied by \(G_t(z)\), the author does not mention many known (and classical!) approaches in this direction. For instance, the author’s favorite example of Apéry’s sequence \(a_n=\sum_{k=0}^n{\binom nk}^2{\binom{n+k}k}^2\) is treated (without addressing the differential equation for \(G(z)=\sum_{n=0}^\infty a_nz^n\)) in [Yu.V. Nesterenko, Math. Notes 59, No. 6, 625–636 (1996; Zbl 0888.11028)], and the application of the saddle-point method as in Nesterenko’s paper could be used in more general settings.
Let the reviewer conclude with another simple method which admits an extension to the general balanced terms considered by the author, on an example of the sequence \(a_n=\sum_{k=0}^n\binom nk{\binom{n+k}k}^2(-1/x)^k\) for \(0<x\leq1\). Note that the asymptotics of \(a_n\) is computed by the ‘practical’ approach in [W. Zudilin, J. Comput. Appl. Math. 202, No. 2, 450–459 (2007; Zbl 1220.65028)] and that specialization \(x=1\) gives one the sequence of numbers used by R. Apéry in his proof of the irrationality of \(\zeta(2)\). Since \[ \begin{gathered} -\frac1{(\xi-1)^{n+1}} =\sum_{\nu=0}^\infty\frac{\nu(\nu-1)\dotsb(\nu-n+1)}{n!}\xi^{\nu-n} =\sum_{k=0}^\infty\binom{n+k}k\xi^k, \\ \biggl(1-\frac\xi x\biggr)^n =\sum_{k=0}^n\binom nk\biggl(-\frac1x\biggr)^k\xi^k, \end{gathered} \] the number \(a_n\) is equal to the constant term of the expression \[ F(\xi_1,\xi_2) =\biggl(1-\frac1{x\xi_1\xi_2}\biggr)^n \frac1{(1-\xi_1)^{n+1}}\,\frac1{(1-\xi_2)^{n+1}}. \] By Cauchy’s integral theorem \[ a_n=\frac1{(2\pi i)^2}\int_{|\xi_1|=c_1}\int_{|\xi_2|=c_2} F(\xi_1,\xi_2)\,\frac{\roman d\xi_1}{\xi_1}\,\frac{\roman d\xi_2}{\xi_2}, \] where radii \(c_1,c_2\) of the circles may take any values in the intervals \(0<c_1<1\) and \(0<c_2<1\). Writing the integral as \[ a_n=\frac1{(2\pi i)^2}\int_{|\xi_1|=c_1}\int_{|\xi_2|=c_2} e^{nf(\xi_1,\xi_2)}\,\frac{\roman d\xi_1}{\xi_1(1-\xi_1)} \,\frac{\roman d\xi_2}{\xi_2(1-\xi_2)}, \] where \[ f(\xi_1,\xi_2) =\log\biggl(1-\frac1{x\xi_1\xi_2}\biggr) -\log(1-\xi_1)-\log(1-\xi_2), \] we are faced to application of the saddle-point method. The candidates for saddle points satisfy \[ \begin{aligned} \frac{\partial f}{\partial\xi_1}(\xi_1,\xi_2) &=\frac1{1-\xi_1}-\frac1{\xi_1(1-x\xi_1\xi_2)}=0, \\ \frac{\partial f}{\partial\xi_1}(\xi_1,\xi_2) &=\frac1{1-\xi_2}-\frac1{\xi_2(1-x\xi_1\xi_2)}=0, \end{aligned} \] and from this system we derive that \(\xi_1=\xi_2=\xi\) with \(\xi\) such that \[ \frac1{1-\xi}-\frac1{\xi(1-x\xi^2)}=0 \] or, equivalently, \(x\xi^3-2\xi+1=0\). The polynomial \(x\xi^3-2\xi+1\) has three real zeros located in the intervals \(\xi<-1\), \(0<\xi<1\) and \(\xi>1\). Taking the zero \(\xi^0\) in the interval \(0<\xi<1\) and choosing \(c_1=c_2=\xi^0\) in the above integral, we see that the \(\xi_1=\xi^0\) and \(\xi_2=\xi^0\) is the unique maximum of \(\text{Re}f(\xi_1,\xi_2)\) on the contour of integration. Therefore, \[ \lim_{n\to\infty}\frac{\log|a_n|}n =\text{Re}f(\xi^0,\xi^0) =\log\biggl(\frac1{x(\xi^0)^2}-1\biggr)-2\log(1-\xi^0). \] {}From \(x\xi^3-2\xi+1=0\) for \(\xi=\xi^0\) we have \(x(\xi^0)^2=(2\xi^0-1)/\xi^0\), whence the right-hand side of the last expression may be simplified: \[ \lim_{n\to\infty}\frac{\log|a_n|}n =\log\frac1{(1-\xi^0)(2\xi^0-1)}. \] Finally, observe that the roots of the polynomial \(x\xi^3-2\xi+1\) are mapped by \[ \xi\mapsto\lambda=e^{f(\xi,\xi)}=\biggl(\frac1{x\xi^2}-1\biggr)\frac1{(1-\xi)^2} \] to the roots of the polynomial \[ x(x-1)\lambda^3-(3x^2-20x+16)\lambda^2+x(3x+8)\lambda-x^2, \] and the latter roots together with \(\lambda=0\) form the corresponding set \(\Sigma_t\) in this case.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
05A16 Asymptotic enumeration
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C70 Other hypergeometric functions and integrals in several variables
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain

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