F. Jouhet
We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey's celebrated ${}_6ψ_6$ summation formula, answering a question recently raised by Chen and Fu ({\em Semi-Finite Forms of Bilateral Basic Hypergeometric Series}, Proc. Amer. Math. Soc., to appear).
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We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.
In this paper, we prove several transformation formulas for the very-well-poised bilateral basic hypergeometric $_5ψ_5$ series by using the relationship between the bilateral basic hypergeometric $_5ψ_5$ series and basic hypergeometric $_8ϕ_7$ series.
In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8ψ_8$ series and two $_8ϕ_7$ series.
Theta functions and transformations of bilateral basic hypergeometric series
本文通过theta函数和双侧基本超几何级数建立新变换公式，构造伴生q级数，证明McIntosh关于q级数渐近变换的猜想，并扩展了Ramanujan和McIntosh的一些恒等式。
We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …
A $q$-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series
本文提出一种基于一阶q^{±1}导数的q-指数算子，用于推导多种双边基本超几何级数的求和公式。
A family of orthogonal functions on the unit circle and a new multilateral matrix inverse
本文利用Bailey求和证明了特定级数构成单位圆上的正交函数族，并从Dougall求和提取出新的双边矩阵逆，结合Pfaff-Saalschütz求和推导了 $_3H_3$ 级数求和公式及其多变量扩展。
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities
本文通过双侧双和罗杰斯-拉马努金恒等式建立新恒等式，并利用基本超几何级数理论和积分方法证明，推导出多种新的多重和罗杰斯-拉马努金型恒等式。
The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type g_n. Our main results include a new q-binomial theorem, new q-Gauss sum, and several transformation formulae for g_n series.
We give new proofs for certain bilateral basic hypergeometric summation formulas using the symmetries of the corresponding series. In particular, we present a proof for Bailey's $_3ψ_3$ summation formula as an application. We also prove a multiple series analogue of this identity by considering hyperoctahedral group symmetries of higher ranks.
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.
A one-parameter generalisation R_λ(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for R_λ(X;b). We also prove a new multiple q-Gauss summation formula and several further results for sl_n basic hypergeometric series based on R_λ(X;b).
We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit evaluations for such multiple series and integrals. We concentrate on such results which do not directly extend to the elliptic level. This text is a provisional version of a chapter on hypergeometric and basic hypergeometric series and integrals associated with root systems for volume 2 of the new Askey--Bateman project which deals with "Multivariable special f…
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…
Factorization of Basic Hypergeometric Series
本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We employ a one-variable extension of q-rook theory to give combinatorial proofs of some basic hypergeometric summations, including the q-Pfaff-Saalschütz summation and a 4phi3 summation by Jain.
A remarkable basic hypergeometric identity
本文给出当基q取单位根时截断基本超几何级数之商的闭形式，贡献了一个新的基本超几何恒等式。
Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7 summations and other identities. We also use similar methods to derive new multivariable 1-psi-1 summations. Some of our results are extended to the case of elliptic hypergeometric series.
Some $q$-supercongruences for multiple basic hypergeometric series
该研究利用超几何级数公式、中国剩余定理等工具，建立了多重基本超几何级数在分圆多项式五、六次幂模下的$q$-超同余式，推广了已有结果并补充了新结论。
We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.
New $q$-identities Via $q$-Derivative of Basic Hypergeometric Series with Respect to Parameters
本文通过$q$-微分和变形$q$-指数算子作用于基本超几何级数，从已知求和与变换公式推导出新的$q$-恒等式。
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$π$ and other eleven nice $π$-formulas. Meanwhile, $q$-analogues of three beautiful series for $ζ(3)$ are also given in the same way.
We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the $q$-Zeilberger algorithm. This method applies to almost all nonterminating basic hypergeometric summation formulas in the book of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulas, including the Sears-Carlitz transformation, transformations of the very-well-poised $_8ϕ_7$ series, t…
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.
We give a Newton type rational interpolation formula (Theorem \ref{theo}). It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu, which allows to recover many important classical $q$-series identities. We show in particular that some bibasic identities are a consequence of our formula.
By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4ϕ_3$ summation, Carlitz's $_5ϕ_4$ summation, Sears' $_3ϕ_2$ to $_5ϕ_4$ transformation, Sears' ${}_4ϕ_3$ transformations, Chen's bibasic summation, Gasper's split poised $_{10}ϕ_9$ transformation, Chu's bibasic symmetric transformation. Along the same line, finite forms of Sylvester's identity, Jacobi's triple product identity, and Kang's identity are also obtained.
We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $Φ_n(q)^2$, where $Φ_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward some conjectures on $q$-congruences modulo $Φ_n(q)^2$.
Expansions of averaged truncations of basic hypergeometric series
受Andrews和Merca工作的启发，本文为更一般的基本超几何级数的平均截断建立展开，涵盖雅可比三重积、$q$-高斯及${}_5ϕ_5$求和，并恢复现有结果、导出新不等式。
With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].
We give a formula for the cyclotomic valuation of $q$-Pochhammer symbols in terms of (generalized) Dwork maps. We also obtain a criterion for the $q$-integrality of basic hypergeometric series in terms of certain step functions, which generalize Christol step functions. This provides suitable $q$-analogs of two results proved by Christol: a formula for the $p$-adic valuation of Pochhammer symbols and a criterion for the $N$-integrality of hypergeometric series.
With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger's "holonomic systems approach" in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action introduce to the basic techniques. An application highlight is a new computer-assisted proof of the celebrated Ismail-Zhang formula, an important q-analog of a classical expansion formula of plane waves …
We provide several new $q$-congruences for truncated basic hypergeometric series with the base being an even power of $q$. Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing $q$-congruences for truncated basic hypergeometric series with the base being an odd power of $q$. We also give a number of related conjectures including $q$-congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a pr…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2ϕ_1$ transformation formula and Sears' ${}_3ϕ_2$ transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is als…
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6-phi-5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8-phi-7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexe…
A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl_3 Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl_3 series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl_3 Selberg integral for Jack p…
We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.
We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the…
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。
It is conjectured that a class of n-fold integral transformations {I(alpha)|alpha in {C}} forms a mutually commutative family, namely, we have I(alpha) I(beta)=I(beta) I(alpha) for all alpha, beta in {C}. The commutativity of I(alpha) for the two-fold integral case is proved by using several summation and transformation formulas for the basic hypergeometric series. An explicit formula for the complete system of the eigenfunctions for n=3 is conjectured. In this formula and in a partial result for n=4, it is observed that all the eigenfunctions do not depend on the spectral parameter alpha of I…
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.
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include $q$-analogues of supercongruences (referring to $p$-adic identities remaining valid for some higher power of $p$) established by Long, by Long and Ramakrishna, and several other $q$-supercongruences. The six basic hypergeometric transformation formulas which ar…
With the use of the $(f,g)$-matrix inversion under specializations that $f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula leads us to some expansion formulas expressing any ${}_{r}ϕ_{s}$ series in variable $x~t$ in terms of a linear combination of ${}_{r+2}ϕ_{s+1}$ series in $t$, as well as various specifications. All these results can be regarded as common generalizations of many konwn expansion formulas in the setting of $q$-series. As specific applications, some new transformation formulas of …
Positivity of the basic hypergeometric series and the MacMahon identity
本文研究满足特定约束的基本超几何级数归一化后的正多项式，给出其三个公式：MacMahon 恒等式的推广、多重集排列统计量的生成函数及有限和表达式。
A series of conjectures is obtained as further investigation of the integral transformation I(alpha) introduced in the previous paper. A Macdonald-type difference operator D is introduced. It is conjectured that D and I(alpha) are commutative with each other. Studying the series for the eigenfunctions under termination conditions, it is observed that a deformed Weyl group action appears as a hidden symmetry. An infinite product formula for the eigenfunction is found for a spacial case of parameters. A one parameter family of hypergeometric-type series F(alpha) is introduced. The series F(alpha…
We give multidimensional generalizations of several transformation formulae for basic hypergeometric series of a specific type. Most of the upper parameters of the series differ multiplicatively from corresponding lower parameters by a nonnegative integer power of the base q. In one dimension, formulae for such series have been found, in the q -> 1 case, by B. M. Minton and P. W. Karlsson, and in the basic case by G. Gasper, by W. C. Chu, and more recently by the author. Our identities involve multilateral basic hypergeometric series associated to the root system A_r (or equivalently, the unit…
In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…
With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $π$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.
Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.
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