Generalized hypergeometric functions: product identities and weighted norm inequalities
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Generalized hypergeometric functions: product identities and weighted norm inequalities Arcadii Z. Grinshpan Dedicated to Mourad Ismail and Dennis Stanton Received: 27 September 2011 / Accepted: 30 March 2013 / Published online: 18 April 2013 © Springer Science+Business Media New York 2013
Abstract We describe a method of obtaining weighted norm inequalities for generalized hypergeometric functions. This method is based upon our recent convolution theorem and some classical hypergeometric identities. In particular, it is shown that some product identities involving the divergent hypergeometric series lead to the convergent hypergeometric inequalities. A number of the new weighted norm inequalities for the Gaussian hypergeometric function, confluent hypergeometric function, and other generalized hypergeometric functions are presented. Keywords Generalized hypergeometric series · Hypergeometric identities · Convolutions · Weighted norm inequalities · Gaussian hypergeometric function · Confluent hypergeometric function Mathematics Subject Classification (2000) 33C20 · 26D15 · 44A35 · 33B15 · 30B10
1 Introduction The generalized hypergeometric functions, i.e., functions which are given by the generalized hypergeometric series, play a fundamental role in analysis, mathematical physics, statistics, number theory, and other fields. Many of their properties have been widely known and actively used for years (see, e.g., [1–3, 5, 8, 9, 20, 22–25]). In fact, most of the functions which are usually used in applied mathematics, physics, and engineering can be presented in the form of a convergent hypergeometric series. We mention that the role of the divergent hypergeometric series and the corresponding formal functions is much less visible in applications. One can say the same about the A.Z. Grinshpan () Department of Mathematics and Statistics, University of South Florida, Tampa, Fl 33620, USA e-mail: [email protected]
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A.Z. Grinshpan
role of the sharp inequalities in terms of the generalized hypergeometric functions: physicists, engineers, and even mathematicians used to have only rough estimates at their disposal. In particular, it concerns the weighted norm inequalities of a different kind. Indeed, the standard tools used in this connection are not sensitive enough to the hypergeometric structure and they lead to the analysis of numerous special cases. In this paper, we describe a method which allows us to deal with the hypergeometric peculiarity and obtain various weighted norm inequalities for the generalized hypergeometric functions. This method is essentially based upon our recent convolution theorem [14] and some classical hypergeometric identities. In particular, we show that some product identities involving the divergent hypergeometric series lead to the convergent hypergeometric inequalities. A number of the new weighted norm inequalities for the Gaussian hypergeometric function, confluent hypergeometric function, and other generalized hypergeometric functions are presented in Sect. 4. In Sect. 2, we overvie
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