REMARKS ON RECURRENCE FORMULAS FOR THE APOSTOL-TYPE NUMBERS AND POLYNOMIALS
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In this paper, by differentiating the generating functions for one of the family of the Apostol-type numbers and polynomials with re- spect to their parameters, we present some partial differential equations including these functions. By making use of these equations, we provide some new formulas, relations and identities including these numbers and polynomials and their derivatives. Furthermore, by using a col- lection of the generating functions for the aforementioned family and their functional equations, we investigate the numbers and polynomials belonging to this family and their relationships with other well-known special numbers and polynomials including the Apostol-Bernoulli num- bers and polynomials of higher order, the Apostol-Euler numbers and polynomials of higher order, the Frobenius-Euler numbers and polyno- mials of higher order, the λ-array polynomials, the λ-Stirling numbers, and the λ-Bernoulli numbers and polynomials.
Abstract
In this paper, by differentiating the generating functions for one of the family of the Apostol-type numbers and polynomials with re- spect to their parameters, we present some partial differential equations including these functions. By making use of these equations, we provide some new formulas, relations and identities including these numbers and polynomials and their derivatives. Furthermore, by using a col- lection of the generating functions for the aforementioned family and their functional equations, we investigate the numbers and polynomials belonging to this family and their relationships with other well-known special numbers and polynomials including the Apostol-Bernoulli num- bers and polynomials of higher order, the Apostol-Euler numbers and polynomials of higher order, the Frobenius-Euler numbers and polyno- mials of higher order, the λ-array polynomials, the λ-Stirling numbers, and the λ-Bernoulli numbers and polynomials.
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