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Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions. (English) Zbl 1499.05020

Summary: The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

MSC:

05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions
11A25 Arithmetic functions; related numbers; inversion formulas
11B37 Recurrences
11B68 Bernoulli and Euler numbers and polynomials
11B83 Special sequences and polynomials
26C99 Polynomials, rational functions in real analysis
34A99 General theory for ordinary differential equations
33B15 Gamma, beta and polygamma functions
65Q20 Numerical methods for functional equations
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References:

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