Using integrals of squares of certain real-valued special functions to prove that the P\'olya \Xi^*(z) function, the functions K_{iz}(a), a > 0, and some other entire functions have only real zeros

Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_\alpha(z) when \alpha \ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^\alpha(z) when \alpha \ge -2, Jacobi polynomials P_n^{(\alpha,\beta)}(z) when \alpha \ge -1 and \beta \ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojevi\'c and C. V. Stanojevi\'c, eds., Marcel Dekker, Inc., 1994, 171--186.], integrals of squares of certain real-valued special functions are used to prove the reality of the zeros of the P\'olya \Xi^*(z) function, the K_{iz}(a) functions when a > 0, and some other entire functions.Comment: 8 page

Lecture notes for an introductory minicourse on q-series

These lecture notes were written for a mini-course that was designed to introduce students and researchers to {\it $q$-series,} which are also called {\it basic hypergeometric series} because of the parameter $q$ that is used as a base in series that are ``{\it over, above or beyond}'' the {\it geometric series}. We start by considering $q$-extensions (also called $q$-analogues) of the binomial theorem, the exponential and gamma functions, and of the beta function and beta integral, and then progress on to the derivations of rather general summation, transformation, and expansion formulas, integral representations, and applications. Our main emphasis is on methods that can be used to {\bf derive} formulas, rather than to just {\it verify} previously derived formulas

Using sums of squares to prove that certain entire functions have only real zeros

It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$ when $\alpha \ge -1,$ confluent hypergeometric functions {}_0F_1(c\/; z) when $c>0$ or $0>c>-1$, Laguerre polynomials $L_n^\alpha(z)$ when $\alpha \ge -2,$ and Jacobi polynomials $P_n^{(\alpha,\beta)}(z)$ when $\alpha \ge -1$ and $\beta \ge -1.$ Besides yielding new inequalities for $|F(z)|^2,$ where $F(z)$ is one of these functions, the derived identities lead to inequalities for $\partial |F(z)|^2/\partial y$ and $\partial ^2 |F(z)|^2/\partial y^2,$ which also give new proofs of the reality of the zeros

Elementary derivations of summations and transformation formulas for q-series

We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric Series" book (which we will refer to as BHS), the Bailey [1935] and Slater [1966] books, and in some papers; thus providing deeper insights into the theory of q-series. Our main emphasis is on methods that can be used to derive formulas, rather than to just verify previously derived or conjectured formulas. In section 5 this approach leads to the derivation of a new family of summation formulas for very well poised basic hypergeometric series _{6+2k}W_{5+2k}, k = 1,2,.... Several of the observations in this paper were presented, along with related exercises, in the author's minicourse on "q-Series" at the Fields Institute miniprogram on "Special functions, q-Series and Related Topics," June 12-14, 1995

q-Analogues of Some Multivariable Biorthogonal Polynomials

A q-analogue a pair of multivariable biorthogonal polynomials found by M.V.Tratnik in 1989 is derived. The weight function is a product of a multivariable version of the integrand in the Askey-Roy integral and of the Askey-Wilson weight function in a single variable. In addition, a biorthogonality relation is derived for certain bivariate extensions of the $q$-Racah polynomials.Comment: 16 pages. To appear in Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman, M. E. H. Ismail and E. Koelink, eds., Dev. Math., Kluwer Acad. Publ., Dordrech

Some curious q-series expansions and beta integral evaluations

We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some curious beta-type integral evaluations which appear to be new.Comment: 14 pages, dedicated to Dick Aske

Summation, transformation, and expansion formulas for multibasic theta hypergeometric series

After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the identities presented here generalize corresponding formulas given in Chapter 11 of the Gasper and Rahman book [Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics And Its Applications 96, Cambridge University Press, Cambridge, 2004].Comment: 18 pages; for the proceedings of the workshop "Elliptic Integrable Systems" (RIMS, Kyoto, November 8-11, 2004), special volume of the Rokko Lectures in Mathematic

Some Systems of Multivariable Orthogonal q-Racah polynomials

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.Comment: 15 page

The Sums of a Double Hypergeometric Series and of the First m+1 Terms of 3F2(a,b,c;(a+b+1)/2,2c;1) when c = -m is a Negative Integer

A summation formula is derived for the sum of the first m+1 terms of the 3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This summation formula is used to derive a formula for the sum of a terminating double hypergeometric series that arose in another project by one of us (C.D.)Comment: 7 pages, added general remarks, another derivation of one of the formulas, and references to related results in basic hypergeometric serie

Errata, updates of the references, etc., for the book Basic Hypergeometric Series

Here are the latest errata, etc., to the Gasper and Rahman "Basic Hypergeometric Series" book. Any additional errata will be added to the end of the last list.Comment: Additional errata and material adde