A Note on the Bailey Transform, the Bailey Pair and the WP Bailey Pair and Their Applications

ver the years, the study of Bailey transform, Bailey lemma, Bailey pair, their variants and their applications are the major subjects of interest. Of course, it is due to the efficiency of the Bailey transform and lemma in producing many ordinary and q-hypergeometric identities, multiple series summation and transformation formulas, and the Rogers-Ramanujan type identities. Andrews investigated a WP-Bailey lemma and the pairs with the help of Bailey transform and used it to derive well-known summations and multiple series transformations. In this research paper, we investigate an Andrews’ type WP-Bailey lemma and the pairs with the help of First Bailey Type Transform due to Joshi and Vyas. The investigated Andrews’ type WP-Bailey lemma is then applied to obtain terminating multiple q-hypergeometric identities and construct the WP-Bailey type chains and a binary tree. The paper is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including number theory, theory of partitions and combinatorial analysis as well as in the study of combinatorial generating functions.

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q-gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q-analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q-Hadamard expansion. Expansions and asymptotics for modified q-Bessel functions are derived as special cases from the general expansion.

Analytic number theory is a branch of number theory which inherits methods from mathematical analysis in order to solve difficult problems about the integers [...]

As we have seen in Chapter 13, representations of Chevalley groups lead to q-orthogonal polynomials with positive integral q. Basic hypergeometric functions and q-orthogonal polynomials for arbitrary (including complex) values of q are connected with quantum algebras and groups. They are considered in the present Chapter.

This paper introduces the Spectral Riemann Conjecture (SRC), a novel reformulation of the classical Riemann Hypothesis (RH) through the lens of spectral breaking theory. The Riemann zeta function ζ(s), central to analytic number theory, possesses trivial zeros at negative even integers and non-trivial zeros conjectured to lie on the critical line Re(s) = 1/2. Building on the verified simplicity of the first 109 non-trivial zeros, the SRC proposes a two-step transformation: decoherence (D), which projects all non-trivial zeros onto Re(s) = 1/2, creating a zero of infinite multiplicity; followed by renormalization (R), which regularizes this multiplicity to one. The resulting function, spectral(s), resides in the spectral space—a domain of functions with real and simple zeros. The paper rigorously defines the operators D and R, demonstrates their action on ζ(s), and proves the convergence of spectral(s) using zeta and Hurwitz regularization techniques. Numerical simulations validate the SRC by approximating ζ(s) with finite products and applying D and R to yield spectral(s), whose zeros are located at {-2, -4, ..., 1/2}. This transformation preserves the infinite zeros, maintaining the arithmetic essence of ζ(s) while simplifying its non-trivial structure. The SRC offers a fresh perspective on RH, aligning with physical regularization analogies and suggesting potential applications in prime number distribution. Future work may extend these simulations and explore deeper implications within analytic number theory and mathematical physics.

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q q -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2 Φ 1 ( q − n , q b + 1 ; q − c + b − n ; q , q − c + d − 1 z ) } n = 0 ∞ \{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty } , where 0 > q > 1 0 > q > 1 and the complex parameters b b , c c and d d are such that b ≠ − 1 , − 2 , … b \neq -1, -2, \ldots , c − b + 1 ≠ − 1 , − 2 , … c-b+1 \neq -1, -2, \ldots , R e ( d ) > 0 \mathcal {R}e(d) > 0 and R e ( c − d + 2 ) > 0 \mathcal {R}e(c-d+2) > 0 . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

This work is motivated essentially by the fact that the applications of basic (or q-) hypergeometric functions are frequently needed in the form of summations, transformations, expansions, reductions, and integral formulas. The objective of this research paper is to study the applications of the general summation formulas contiguous to q-Kummer theorems in deriving several presumably new q-integral evaluations involving the basic (or q-) hypergeometric functions of type $$_{3} \Phi_{3}$$ and $$_{3} \Phi_{2}$$ . The results are further particularized to give specific integral evaluations in the form of summations to show the importance and utilizations of our main findings. We also derive the summation theorems for certain basic (or q-) Appell double hypergeometric functions, basic (or q-) Lauricella multiple hypergeometric functions, and certain basic (or q-) Laplace transforms by applying general summation formulas contiguous to q-Kummer theorems.

Since the deep paper by Bohr and Kalckar in 1938, it has been known that the Ramanujan formula in number theory is related to statistical physics and nuclear theory. From the early 1970s, there have been attempts to generalize number theory from the space of integers to the space of rational numbers, i.e., to construct a so-called analytic number theory. In statistical physics, we consider parameters such as the volume V, temperature T, and chemical potential μ, which are not integers and are consequently related to analytic number theory. This relation to physical concepts leads us to seek new relations in analytic number theory, and these relations turn out to be useful in statistical physics.

The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series. We begin with discussion of some of the lesser known identities of L.J. Rogers which have interesting interpretations in the theory of partitions. Illustrations are given for the numerous ways partition studies lead to Eulerian series. The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions. Perhaps it is most interesting to note that we deduce the Rogers-Ramanujan identities from our solution to the connection coefficient problem for the little q-Jacobi polynomials.

Summary. The computational solution of a certain class of differential-difference equations requires numerical procedures involving an extremely high degree of precision to obtain accurate results over a large range of the independent variable. One method of solution uses an iterative procedure which relates the differentialdifference equation over a large range to a system of ordinary differential equations over a limited range. When the characteristic roots of the related system indicate borderline stability, it is evident that small perturbations in obtaining successive initial values eventually grow out of control as the system increases. To investigate this phenomenon, we examine the equation u' (x) =-u (x -1) /x. arising in analytic number theory.

Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

Darling [2] in 1932 gave two types (equations 11 and 18) of transformations connecting generalised hyper geometric functions. The first was studied by Bailey [1] and extended by Sears [4] to a transformation connecting products of basic hyper geometric functions of the type r+1ϕr × r+1ϕr. In a number of papers [6, 7, 8] the author has extended these results to both unilateral and bilateral series with bases q and q1/2. The second type of transformation by Darling for a product 0F1 × 3F2 was extended by Bailey [1] to a transformation between 1F0 × r+1Fr. In the same paper Bailey mentioned the transformation of a 0ϕ1 × 3ϕ2 without proof.

We begin with the duality between analytic number theory, combinatorial identities and q-series, to indicate the historical development of the allied disciplines. It is irrelevant what notation we use for the Γ-function, the essential part is that we keep this notation. Section 3.7 is devoted to this important function and the hypergeometric function. We use a vector notation for the Γ-function and introduce the concepts well-poised and balanced series. The binomial coefficients also play an important part since a finite hypergeometric series can always be expressed in two equivalent ways. The three Kummerian summation formulae (and their multiple q-analogues) will follow us in future chapters. We summarise the different schools for Theta functions and show that the elliptic function snu can be written as a balanced quotient of infinite q-shifted factorials. We conclude this chapter with definitions of the most important orthogonal polynomials; we keep Jacobi’s definition for the Jacobi polynomials.

We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of zeroes of the Riemann zeta-function. These results were obtained by creating the Analytic Number Theory Exponent Database to collect results and relationships between these exponents, and then systematically optimising these relationships to obtain the new bounds. We welcome further contributions to the database, which aims to allow easy conversion of new bounds on these exponents into optimised bounds on other related exponents of interest.