The aim of this research note is to prove the following new transformation formula begin{equation*} (1-x)^{-2a},_{3}F_{2}left[begin{array}{ccccc} a, & a+frac{1}{2}, & d+1 & & \ & & & ; & -frac{4x}{(1-x)^{2}} \ & c+1, & d & & end{array}right] \ =,_{4}F_{3}left[begin{array}{cccccc} 2a, & 2a-c, & a-A+1, & a+A+1 & & \ & & & & ; & -x \ & c+1, & a-A, & a+A & & end{array} right], end{equation*} wher...

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

The main purpose of this paper is to compute all irreducible spherical functions on G = SL(2,C) of arbitrary type δ ∈ K̂, where K = SU(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the three dimensional hyperbolic space H = G/K. The entries of H are solutions of two coupled systems of ordinary differential equations. By an appropriate twisti...

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in ...

We derive relations for a certain class of terminating 4F3(4) hypergeometric series with three free parameters. The invariance group composed these is shown to be isomorphic the symmetric S3. further study 3F2(4) that fall under two families. By using reversal, we examine corresponding 4F3(1/4) and 3F2(1/4) relations. additionally formulas sums first n + 1 terms several nonterminating series. a...

The goal of this study is to develop some new connection formulae between two generalized classes Fibonacci and Lucas polynomials. Hypergeometric functions the kind 2F1(z) are included in all coefficients for a specific z. Several famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, Fermat–Lucas deduced special cases derived formulae. Some introduced generalize those existing...

In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms an...

Following suggestions of T. H. Koornwinder [3], we give a new proof of Kummer’s theorem involving Zeilberger’s algorithm, the WZ method and asymptotic estimates. In the first section, we recall a classical proof given by L. J. Slater [7]. The second section discusses the new proof, in the third section sketches of similar proofs for Bailey’s and Dixon’s theorems are given. The author is gratefu...

∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the ...

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families rational functions in three variables x, y, z, using creative telescoping, yielding modular forms expressed as pullbacked 2F1 hypergeometric functions, can be much more efficiently by calculating j-invariant an elliptic curve canonically associated with denominator functions. These drastically gener...