arXiv subjects

S. I. Kalmykov

Publications and source records attributed to S. I. Kalmykov.

7 recordsLinked to original sources

arXiv2021

A new identity for the sum of products of generalized basic hypergeometric functions

We prove a duality relation for generalized basic hypergeometric functions. It forms a $q$-extension of a recent result of the second and the third named authors and generalizes both a $q$-hypergeometric identity due to the third named author (jointly with Feng and Yang) and a recent identity for the Heine's ${}_2ϕ_{1}$ function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furnish a number of explicit examples.

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arXiv2018

Inequalities for some basic hypergeometric functions

We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric function with respect simultaneous shift of all its parameters. For a particular case of Heine's basic hypergeometric function we prove logarithmic concavity and convexity with respect to the bottom parameter. We further establish a linearization identity for the generalized Turánian formed by a particular case of Heine's basic hypergeometric function. Its $q=1$ case also appears to be new.

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arXiv2017

New identities for a sum of products of the Kummer functions

Recently, Feng, Kuznetsov and Yang discovered a very general reduction formula for a sum of products of the generalized hypergeometric functions (J. Math. Anal. Appl. 443(2016), 116--122). The main goal of this note is to present a generalization of a particular case of their identity when the generalized hypergeometric function is reduced to the Kummer function 1F1. Our generalized formula contains an additional integer shift in the bottom parameter of the Kummer function. The key ingredient of the proof is a summation formula for the Clausen series 3F2(1) with two integral parameter differences of opposite sign. In the ultimate section of the paper we prove another formula for a particular product difference of the Kummer functions in terms of a linear combination of these functions.

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arXiv2017

Inequalities for series in q-shifted factorials and q-gamma functions

The paper studies logarithmic convexity and concavity of power series with coefficients involving q-gamma functions or q-shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such series are basic hypergeometric functions. We consider four types of series. For each type we establish conditions sufficient for the power series coefficients of the generalized Turánian formed by these series to have constant sign. Finally, we furnish seven examples of basic hypergeometric functions satisfying our general theorems. This investigation extends our previous results on power series with coefficient involving the ordinary gamma functions and the shifted factorials.

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arXiv2016

Log-concavity and Turán-type inequalities for the generalized hypergeometric function

The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to simultaneous shift of several parameters. We use integral representations and properties of Meijer's $G$ function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Turánian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a conjecture about reality of zeros.

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arXiv2012

Log-convexity and log-concavity for series in gamma ratios and applications

Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of $q$-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.

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arXiv2012

Log-concavity for series in reciprocal gamma functions and applications

Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series in reciprocal gamma functions, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave if the sequence of coefficients times factorial is log-concave and the sum is discrete Wright log-concave if the coefficents are log-concave. We conjecture that the latter condition is in fact sufficient for the log-concavity of the sum. We exemplify our general theorems by deriving known and new inequalities for the modified Bessel, Kummer and generalized hypergeometric functions and their parameter derivatives.

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