Abstract

The classical power means of Hardy, Littlewood and Polya, which contains the harmonic mean, arithmetic mean and geometric mean, is generalized to the $Y$-mean and hypergeometric mean by Carlson. Carlson's hypergeometric mean is to average a function over a type-1 Dirichlet measure, and this term in the current literature is known as the Dirichlet average of that function. The present paper introduces a new Dirichlet average, associated with the modified type-1 Dirichlet measures called modified type-1 Dirichlet averages. This paper also investigates the modified type-1 Dirichlet averages of a three-parameter Mittag-Leffler type function, which is expressed using Riemann-Liouville integrals and hypergeometric functions with multiple variables.

Keywords and Phrases

Dirichlet average, generalized type-1 and type-2 Dirichlet models, Mittag-Leffler functions, Riemann-Liouville fractional integrals, hyper-geometric functions of one and many variables.

A.M.S. subject classification

62E15, 60E05, 33E12, 26A33, 33C70, 33C20, 33C65.

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