Abstract
In this paper, using generating functions and combinatorial techniques, we extend Agoh and Dilcher’s quadratic recurrence formula for Bernoulli numbers in (Agoh and Dilcher in J. Number Theory 124:105-122, 2007) to Apostol-Bernoulli and Apostol-Euler polynomials and numbers. MSC: 11B68; 05A19
Highlights
1 Introduction The classical Bernoulli polynomials Bn(x) and Euler polynomials En(x) have played important roles in many branches of mathematics such as number theory, combinatorics, special functions and analysis, and they are usually defined by means of the following generating functions: text tn et – = Bn(x) n!
In this paper, using generating functions and combinatorial techniques, we extend the above mentioned Agoh and Dilcher quadratic recurrence formula for Bernoulli numbers to Apostol-Bernoulli and Apostol-Euler polynomials and numbers
Applying ( . ), ( . ) and ( . ) to ( . ), in view of the Cauchy product and the complementary addition theorem of the Apostol-Bernoulli polynomials, we derive
Summary
Several interesting properties for Apostol-Bernoulli and Apostol-Euler polynomials and numbers have been presented in [ – ]. Differential relations of the Apostol-Bernoulli and Apostol-Euler polynomials of order α: for non-negative integers k and n with ≤ k ≤ n, Difference equations of the Apostol-Bernoulli and Apostol-Euler polynomials of order α: for a positive integer n, λBn(α)(x + ; λ) – Bn(α)(x; λ) = nBn(α–– )(x; λ) Bn( )(x; λ) = xn , Addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials of order α: for a suitable parameter β and a non-negative integer n, n
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