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Identities Involving Generalized Harmonic Numbers and Other Special Combinatorial Sequences

Huyile Liang

Open publisher page 6 citations

Abstract

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Rior-dan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of some summations associ-ated with the numbers Hn,k,r(α, β) by Darboux’s method and Laplace’s method. 1

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What this paper is about

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Rior-dan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of some summations associ-ated with the numbers Hn,k,r(α, β) by Darboux’s method and Laplace’s method. 1

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OpenAlex reports 6 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Rior-dan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of some summations associ-ated with the numbers Hn,k,r(α, β) by Darboux’s method and Laplace’s method. 1

Key concepts: Harmonic number, Bernoulli number, Mathematics, Stirling numbers of the second kind, Stirling numbers of the first kind, Stirling number, Laplace transform, Bernoulli's principle

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