On a generalization of Mittag-Leffler function and its properties

Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

Operators of fractional integration and their applications

Evolution operator equations: Integration with algebraic and finitedifference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory

In the vast literature in Analytic Number Theory, one can find systematic and extensive investigations not only of the classical Bernoulli, Euler and Genocchi polynomials and their corresponding numbers, but also of their many generalizations and basic (or q-) extensions. Our main object in this presentation is to introduce and investigate some of the principal generalizations and unifications of each of these polynomials by means of suitable generating functions. We present several interesting properties of these general polynomial systems including some explicit series representations in terms of the Hurwitz (or generalized) zeta function and the familiar Gauss hypergeometric function. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called -Stirling numbers of the second kind, we derive several properties and formulas and consider some of their interesting applications to the family of the Apostol type polynomials. We also give a brief expository and historial account of the various basic (or q-) extensions of the classical Bernoulli polynomials and numbers, the classical Euler polynomials and numbers, the classical Genocchi polynomials and numbers, and also of their such generalizations as (for example) the above-mentioned families of the Apostol type polynomials and numbers. Relevant connections of the definitions and results presented here with those in earlier as well as forthcoming investigations will be indicated.

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A. W. Kemp

Papers

A treatise on generating functions