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Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials |
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PP: 390-444 |
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Author(s) |
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H. M. Srivastava,
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Abstract |
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| 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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