The aim of this paper is to establish new series transforms of Bailey type and to show that these Bailey type transforms work as efficiently as the classical one and give not only new $q$-hypergeometric identities, converting double or triple series into a very-well-poised $_{10}Φ_9$ or $_{12}Φ_{11}$ series but can also be utilized to derive new double and triple series Rogers-Ramanujan type identities and corresponding infinite families of Rogers-Ramanujan type identities.
We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic $C_{\ell}$ groups. The classical case, corresponding to $A_1$ or equivalently $\roman U(2)$, contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. In particular, our program extends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.
We list $A_n$, $C_n$ and $D_n$ extensions of the elliptic WP Bailey transform and lemma, given for $n=1$ by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced and very-well-poised ${}_{10}V_9$ elliptic hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our study, we discover two new $A_n$ ${}_{12}V_{11}$ transformation formulas, that reduce to two new $A_n$ extensions of Bailey's $_{10}ϕ_9$ transformation formulas when the nome $p$ is $0$, and two multiple series extensions of Frenkel and Turaev's s…
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 obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalschützian series, Saalschützian series to Saalschützian series, and very-well-poised and nearly-poised series to very-well-poised series. We employ a method in which summations and transformations of lower-order series are used to obtain transformations of higher-order series. By taking limits, we also obtain extensions of two classical quadratic transformations of Whipple and Bailey. Furthermore, we show how a number of other well-known results rega…
A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W. N. Bailey in his paper, "Identities of the Rogers-Ramanujan type," [Proc. London Math. Soc. (2), 50 (1949), 421--435]. This leads to the derivation of a number of elegant new Rogers-Ramanujan type identities.
This is a paper published in 2001 based on a talk given in 1999 celebrating the 50th anniversary of W. N. Bailey's influential q-series paper "Identities of the Rogers-Ramanujan type". In no more than 13 pages I give a brief but reasonably complete overview of the mathematics related to Bailey's lemma.
In a recent paper, I defined the "standard multiparameter Bailey pair" (SMPBP) and demonstrated that all of the classical Bailey pairs considered by W.N. Bailey in his famous paper (\textit{Proc. London Math. Soc. (2)}, \textbf{50} (1948), 1--10) arose as special cases of the SMPBP. Additionally, I was able to find a number of new Rogers-Ramanujan type identities. From a given Bailey pair, normally only one or two Rogers-Ramanujan type identities follow immediately. In this present work, I present the set of $q$-difference equations associated with the SMPBP, and use these $q$-difference equat…
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 are interested in extending Bailey's identity to other classical hypergeometric functions. Bailey's identity states that under a suitable choice of parameters, Appell's $F_4$ decomposes into a product of two ${}_2F_1$'s. We will show how Bailey-type factorizations can be found for Horn's hypergeometric functions $H_1, H_4$ and $H_5$.
In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Bas…
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series.
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 …
We present here the $q$-analogues of certain transformations or reduction formulae for Srivastava-Daoust type double hypergeometric series. These reduction formulae are derived by utilizing the extended Bailey's Transform developed and studied by Joshi and Vyas [Int. J. Math. Sci., (12), 2005, 1909-1927]. A number of well-known $q$-hypergeometric transformations are also obtained as special cases of our results.
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
We prove a master theorem for hypergeometric functions of Karlsson-Minton type, stating that a very general multilateral U(n) Karlsson-Minton type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson-Minton summation formula and many of its known generalizations as special cases, and it also implies several "Bailey-type" identities for U(n) hypergeometric series, including multivariable 10-W-9 transformations of Milne and Newcomb and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of mult…
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…
New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type $BC$ and of type $C$. Various transformation and summation formulas are derived as special cases to recover some previously known results.
This paper refers to the study of generalized Struve type function. Using generalized Galue type Struve function (GTSF) by Nisar et al. [13], we derive various integral transform, including Euler transform, Laplace transform, Whittakar transform, K-transform and Fractional Fourier transform. The results are expressed in terms of generalized Wright function. The transform established here are general in nature and are likely to find useful in applied problem of Sciences, engineering and technology. Certain interesting special cases of the main results are also considered.
We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p-L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel or $\mathscr{H}_α$ transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.
New index transforms with Weber type kernels, consisting of products of Bessel functions of the first and second kind are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundary value problem on the wedge for a fourth order partial differential equation.
