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Publications and source records attributed to George Gasper Jr.
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.
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.
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.
A Lemma of Riemann--Lebesgue type for Fourier--Jacobi coefficients is derived. Via integral representations of Dirichlet--Mehler type for Jacobi polynomials its proof directly reduces to the classical Riemann--Lebesgue Lemma for Fourier coefficients. Other proofs are sketched. Analogous results are also derived for Laguerre expansions and for Jacobi transforms.
Sufficient ultraspherical multiplier criteria are refined in such a way that they are comparable with necessary multiplier conditions. Also new necessary conditions for Jacobi multipliers are deduced which, in particular, imply known Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an essential way.
The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier criteria of Stempak and Trebels \cite{ST}. The resulting sufficient $(p,q)$ multiplier criteria are comparable with necessary ones of Gasper and Trebels \cite{laguerre}.
In 1965 K. de Leeuw \cite{deleeuw} proved among other things in the Fourier transform setting: {\it If a continuous function $m(ξ_1, \ldots ,ξ_n)$ on ${\bf R}^n$ generates a bounded transformation on $L^p({\bf R}^n),\; 1\le p \le \infty ,$ then its trace $\tilde{m}(ξ_1, \ldots ,ξ_m)=m(ξ_1, \ldots ,ξ_m,0,\ldots ,0), \; m<n,$ generates a bounded transformation on $L^p({\bf R}^m)$. } In this paper, the analogous problem is discussed in the setting of Laguerre expansions of different orders.
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 $J_α(z)$ when $α\ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when $c>0$ or $0>c>-1$, Laguerre polynomials $L_n^α(z)$ when $α\ge -2,$ and Jacobi polynomials $P_n^{(α,β)}(z)$ when $α\ge -1$ and $ β\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.
The necessary multiplier conditions for Laguerre expansions derived in Gasper and Trebels \cite{laguerre} are supplemented and modified. This allows us to place Markett's Cohen type inequality \cite{cohen} (up to the $\log $--case) in the general framework of necessary conditions.