38 research outputs found

    Identities on the k-ary Lyndon words related to a family of zeta functions

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    The main aim of this paper is to investigate and introduce relations between the numbers of k-ary Lyndon words and unified zeta-type functions which was defined by Ozden et al [15, p. 2785]. Finally, we give some identities on generating functions for the numbers of k-ary Lyndon words and some special numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials, Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.Comment: 9 page

    Interpolation function of the genocchi type polynomials

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    The main purpose of this paper is to construct not only generating functions of the new approach Genocchi type numbers and polynomials but also interpolation function of these numbers and polynomials which are related to a, b, c arbitrary positive real parameters. We prove multiplication theorem of these polynomials. Furthermore, we give some identities and applications associated with these numbers, polynomials and their interpolation functions.Comment: 14 page

    Some results on q-Hermite based hybrid polynomials

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    In this article, a hybrid class of the q-Hermite based Apostol type Frobenius-Euler polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating function methods. Further, the 2D q-Hermite based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials are introduced and important relations for these polynomials are also established. Finally, a new class of the 2D q-Hermite based Appell polynomials is investigated as the generalization of the above polynomials. The determinant definitions for the 2D q-Hermite based Appell and related polynomials are also explored

    Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

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    summary:One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by (2λet+1)αext=n=0En(α)(x;λ)tnn!,λC{1}, \Big ( \frac{2}{\lambda e^t+1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{E}^{(\alpha )}_{n}(x;\lambda ) \frac{t^n}{n!}\,, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace \,, and as an “exceptional family” (tet1)αext=n=0Bn(α)(x)tnn!, \Big ( \frac{t}{e^t-1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{B}^{(\alpha )}_{n}(x) \frac{t^n}{n!}\,, both of these for αC\alpha \in \mathbb{C}
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