R. Milson
The present note considers a certain family of sums indexed by the set of fixed length compositions of a given number. The sums in question cannot be realized as weighted compositions. However they can be be related to the hypergeometric function, thereby allowing one to factorize the corresponding generating polynomials. This factorization leads to some interesting identities.
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A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal Binomial Sums
本文通过逻辑一致性检查、积分推导分析和符号计算，证明Pain提出的参数扩展在递归二项式和中的超几何闭形式表示是错误的。
Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. Additionally, we prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh.
Double basic hypergeometric sums via a regularized Jackson $q$-integral
该研究利用Andrews--Askey积分推导Jackson $q$-积分的全纯延拓公式，将终止恒等式延拓到非终止情形，得到多个基本超几何双和公式与积分表示，关联多个经典变换与生成函数。
$q$-analogs of sum equals integral relations $\sum_{n\in\mathbb{Z}}f(n)=\int_{-\infty}^\infty f(x)dx$ for sinc functions and binomial coefficients are studied. Such analogs are already known in the context of $q$-hypergeometric series. This paper deals with multibasic `fractional' generalizations that are not $q$-hypergeometric functions.
Reciprocal binomial sums via Beta integrals
本文通过Beta积分技术系统地推导了涉及二项式系数倒数的二项式和的求解方法，建立了Frisch公式等经典恒等式，并扩展到参数和，利用超几何函数获得闭式解。
In 1988, Andrews, Dyson and Hickerson initiated the study of q-hypergeometric series whose coefficients are dictated by the arithmetic in real quadratic fields. In this paper, we provide a dozen q-hypergeometric double sums which are generating functions for the number of ideals of a given norm in rings of integers of real quadratic fields and prove some related identities.
Two $q$-Supercongruences Related to Double Sums
本文利用q恒等式和微分算子技术，建立了两个与截断双重基本超几何q级数相关的q超同余式，属于数学领域的同余式研究成果
We represent the sums $\sum_{k=0}^{n-1}{n \choose k}^{-2}$, $\sum_{k=0}^m{m\choose k}^{-1}{a\choose n-k}^{-1}$, $\sum_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\genfrac{[}{]}{0pt}{}{n}{k}}_q}$, and the sum of the reciprocals of the summands in Dixon's identity, each as a product of an {\it indefinite hypergeometric sum} times a (closed form) {\it hypergeometric sequence}
This article studies sums over all compositions of an integer. We derive a generating function for this quantity, and apply it to several special functions, including various generalized Bernoulli numbers. We connect composition sums with a recursive tree introduced by S.G. Woon and extended by P. Fuchs under the name "general PI tree", in which an output sequence $\{x_n\}$ is associated to the input sequence $\{g_n\}$ by summing over each row of the tree built from $\{g_n\}$. Our link with the notion of compositions allows to introduce a modification of Fuchs' tree that takes into account non…
In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of $n$. Such sums appear in the analyses of the Mabinogion urn and the Ehrenfest urn in probability. Using hypergeometric functions, we are able to simplify these sums, and in some cases express them in terms of the harmonic numbers.
We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.
We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values may be obtained. It may be emphasized that we make no reliance on the WZ method for hypergeometric summation. Additional …
By telescoping method, Sun gave some hypergeometric series whose sums are related to $π$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper summability of $r(k)t_k$ for $r(k)$ being a rational function of $k$. We give an upper bound and a lower bound on the degree of the numerator of $r(k)$ such that $r(k)t_k$ is Gosper summable. We also show that the denominator of the $r(k)$ can read from the Gosper representation of $t_{k+1}/t_k$. Based on these results, we give a systematic method to construct series…
There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.
Rational function certification of multisum/integral/``$q$'' identities
将证明终止超几何恒等式的有理函数证明方法从单和或单积分推广到多积分/和及"$q$"积分/和.
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6ψ_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers ${}_6ϕ_5$ and Dougall ${}_5F_4…
We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain a symbolic summation framework that solves this problem for sums defined, e.g., over hypergeometric, $q$-hypergeometric or mixed hypergeometric expressions. Recently, our methods have found applications in quantum field theory.
We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congruences for these sums modulo a prime $p.$ The congruences obtained allow us to find nice $p$-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo $p$. Moreover, we are also able to provide a new proof of Zagier's formula for $ζ^{*}(\{2\}^a,3,\{2\}^b)$ based on a finite…
In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial transform and significantly generalize a previously known result. We also obtain new relations among many classical hypergeometric orthogonal polynomials as well as new formulas for sums involving terminating hypergeometric series.
For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $P^\ast(q)$. We also give a new proof for the number of complete subgraphs of order three contained in $P^\ast(q)$ by evaluating certain characte…
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoretic Dirichlet series.
By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\…
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