In this paper, we construct generating functions for higher-order Euler-type polynomials and numbers. By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type polynomials and numbers. A special case of higher-order Euler-type polynomials is eigenfunctions for the generalized partial Hecke operators. Moreover, we give not only some properties, but also applications for these polynomials and numbers.
The aim of this paper is to give not only the matrix representation of partial Hecke-type operators by means of Bernoulli polynomials and Euler polynomials, but also functional equations and differential equations related to partial Hecke-type operators and special polynomials. By using these functional equations and differential equations, we derive some identities associated with special polynomials and partial Hecke-type operators. Moreover, we find several useful identities and relations using the partial Hecke operators. MSC: 05B20; 11B68; 11F25
In this article, we first determine a sequence { f n (τ)} n∈N of modular forms with weight 2 n k + 4(2 n−1 − 1) (n ∈ N; k ∈ N \ {1}; N := {1, 2, 3, • • • }). We then present some applications of this sequence which are related to the Eisenstein series and the cusp forms. We also prove that higher-order derivatives of the Weierstrass type ℘ 2n-functions are related to the above-mentioned sequence { f n (τ)} n∈N of modular forms.
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