1,256 research outputs found
Relationships Between Generalized Bernoulli Numbers and Polynomials and Generalized Euler Numbers and Polynomials
In this paper, concepts of the generalized Bernoulli and Euler numbers and polynomials are introduced, and some relationships between them are established
Extensions of the Genocchi polynomials and their Fourier expansions and integral representations
Some results for the q-Bernoulli and q-Euler polynomials
AbstractWe show some results for the q-Bernoulli and q-Euler polynomials. The formulas in series of the Carlitz's q-Stirling numbers of the second kind are also considered. The q-analogues of well-known formulas are derived from these results
Bounds for the ratio of two gamma functions--From Wendel's limit to Elezovi\'c-Giordano-Pe\v{c}ari\'c's theorem
In the survey paper, along one of main lines of bounding the ratio of two
gamma functions, we look back and analyse some known results, including
Wendel's, Gurland's, Kazarinoff's, Gautschi's, Watson's, Chu's,
Lazarevi\'c-Lupa\c{s}'s, Kershaw's and Elezovi\'c-Giordano-Pe\v{c}ari\'c's
inequalities, claim, monotonic and convex properties. On the other hand, we
introduce some related advances on the topic of bounding the ratio of two gamma
functions in recent years.Comment: 16 page
Evaluations of the improper integrals ∫ 0 ∞ [sin 2m (αx)]/(x 2n)] cos(bx) dx and ∫ 0 ∞ [sin 2m+1 (αx)]/(x 2n+1 )] cos(bx)dx
In this article, using L’Hospital rule, mathematical induction, the trigonometric power formulae and integration by parts, some integral formulae for improper integrals ∫ 0 ∞ [sin 2m (αx)]/(x 2n)] cos(bx) dx and ∫ 0 ∞ [sin 2m+1 (αx)]/(x 2n+1 )] cos(bx)dx are established, where m ≥ n are all positive integers and real numbers α ≠0 and b ≥ 0
SOME FORMULAS FOR APOSTOL-EULER POLYNOMIALS ASSOCIATED WITH HURWITZ ZETA FUNCTION AT RATIONAL ARGUMENTS
We give some explicit relationships between the Apostol-Euler polynomials and generalized Hurwitz-Lerch Zeta function and obtain some series representations of the Apostol-Euler polynomials of higher order in terms of the generalized Hurwitz-Lerch Zeta function. Several interesting special cases are also shown
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