Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series
R. Jagannathan, K. Srinivasa Rao
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16 pages, To appear in the Proceedings of the International Conference on Number Theory and Mathematical Physics, 20-21 December 2005, Srinivasa Ramanujan Centre, Kumbakonam, India
We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]_{p,q} = (p^n - q^n)/(p-q). This twin-basic number occurs in the theory of two-parameter quantum algebras and has been introduced independently in combinatorics. The (p,q)-identities thus derived, with doubling of the number of parameters, offer more choices for manipulations; for example, results that can be obtained via the limiting process of confluence in the usual q-series framework can be obtained by simpler substitutions. The q-results are of course special cases of the (p,q)-results corresponding to choosing p = 1. This also provides a new look for the q-identities.
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric polynomials and we derive expansion formulae corresponding to all the families within the $q$-Askey scheme.
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
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 $Φ_n(q)^2$.
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].
Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Φ_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
We give some new $q$-supercongruences on truncated forms of squares of basic hypergeometric series. Most of them are modulo the cube of a cyclotomic polynomial, and two of them are modulo the fourth power of a cyclotomic polynomial. The main ingredients of our proofs are the creative microscoping method, a lemma of El Bachraoui, and the Chinese remainder theorem for coprime polynomials. We also propose several related conjectures for further study.
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.
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 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…
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.
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…
The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2ψ_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. \\ New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say $χ(q)$, and …
We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.
If $k$ is set equal to $a q$ in the definition of a WP Bailey pair, \[ β_{n}(a,k) = \sum_{j=0}^{n} \frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}α_{j}(a,k), \] this equation reduces to $β_{n}=\sum_{j=0}^{n}α_{j}$. This seemingly trivial relation connecting the $α_n$'s with the $β_n$'s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for…
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some $q$ orthogonal polynomials, and various multi-sums that are expressible as infinite products.
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.
We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in…
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.
We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WP-Bailey pairs, and use them to derive some additional new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WP-Bailey pairs\\ $(α_n(a,k)$, $β_n(a,k))$, in which $α_n(a,k)$ is independent of $k$, for generalizations of identities of the Rogers-Ramanujan type.
For the hypergeometric series with parameters from the rational fields, there is an effective criterion due to Christol to decide whether the hypergeometric series is N-integral or not. Christol criterion is a basic and vital tool in the recent striking work of Delaygue, Rivoal and Roques on the N-integrality of the hypergeometric mirror maps with rational parameters. In this paper, we develop a systematic theory on the N-integrality of the hypergeometric series with parameters from quadratic fields. We first present a detailed $p$-adic analysis to set up a criterion of the $p$-adic integralit…
