scholarly journals Generalised Bernoulli polynomials and series

2000 ◽  
Vol 61 (2) ◽  
pp. 289-304 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Bessel Function ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Summation Formula

We present several results related to the recently introduced generalised Bernoulli polynomials. Some recurrence relations are given, which permit us to compute efficiently the polynomials in question. The sums , where jk = jk (α) are the zeros of the Bessel function of the first kind of order α, are evaluated in terms of these polynomials. We also study a generalisation of the series appearing in the Euler-MacLaurin summation formula.

scholarly journals Some New Recurrence Relations Concerning Jacobi Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hatem Mejjaoli ◽  
Ahmedou Ould Ahmed Salem
Keyword(s):  
Bessel Function ◽  
Recurrence Relations ◽  
Elementary Functions ◽  
Jacobi Function ◽  
Jacobi Functions ◽  
Asymptotic Relationship

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functionsΦλα,β. Based on an asymptotic relationship between the Jacobi function and the Bessel function, the expression of Bessel function in terms of elementary functions follows as particular cases.

scholarly journals Some Identities on the Generalizedq-Bernoulli,q-Euler, andq-Genocchi Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Daeyeoul Kim ◽  
Burak Kurt ◽  
Veli Kurt
Keyword(s):  
Recurrence Relations ◽  
Analogous Result ◽  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
Euler Polynomials ◽  
Genocchi Polynomials

Mahmudov (2012, 2013) introduced and investigated someq-extensions of theq-Bernoulli polynomialsℬn,qαx,yof orderα, theq-Euler polynomialsℰn,qαx,yof orderα, and theq-Genocchi polynomials𝒢n,qαx,yof orderα. In this paper, we give some identities forℬn,qαx,y,𝒢n,qαx,y, andℰn,qαx,yand the recurrence relations between these polynomials. This is an analogous result to theq-extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).

scholarly journals THE GENERALIZED Q-BESSEL MATRIX FUNCTION OF TWO VARIABLES

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Fadhl S. N. Alsarahi
Keyword(s):  
Difference Equation ◽  
Bessel Function ◽  
Generating Function ◽  
Matrix Function ◽  
Recurrence Relations ◽  
Special Function ◽  
Applied Mathematics ◽  
Function Of Two Variables

The Bessel function is probably the best known special function, within pure and applied mathematics. In this paper, we introduce the generalized q-analogue Bessel matrix function of two variables. Some properties of this function, such as generating function, q-difference equation, and recurrence relations are obtained.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals Some identities and recurrence relations on the two variables Bernoulli, Euler and Genocchi polynomials

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1757-1765
Author(s):  
Veli Kurt ◽  
Burak Kurt
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Addition Theorem ◽  
Euler Polynomials ◽  
Genocchi Polynomials

Mahmudov in ([16], [17], [18]) introduced and investigated some q-extensions of the q-Bernoulli polynomials B(?)n,q (x,y) of order ?, the q-Euler polynomials ?(?)n,q (x,y) of order ? and the q-Genocchi polynomials G(?)n,q (x,y) of order ?. In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and the recurrence relation between these polynomials. We give a different form of the analogue of the Srivastava-Pint?r addition theorem.

scholarly journals Analytic continuation for multiple zeta values using symbolic representations

2019 ◽  
Vol 16 (03) ◽  
pp. 579-602
Author(s):  
Lin Jiu ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Analytic Continuation ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Renormalization Procedure ◽  
Symbolic Representations ◽  
Multiple Zeta Function ◽  
Multiple Zeta ◽  
The Family ◽  
Zeta Values

We introduce a symbolic representation of [Formula: see text]-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber’s formula, and as the result of a natural renormalization procedure for Faulhaber’s formula.

Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter

1958 ◽  
Vol 250 (984) ◽  
pp. 479-517 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Function ◽  
Complex Variable ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Airy Function ◽  
Formal Series ◽  
Complex Parameter ◽  
Second Order Differential Equations ◽  
Uniform Asymptotic Expansions

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,>4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.

scholarly journals Truncated Fubini Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 431 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Summation Formulas

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials.

scholarly journals On Gould–Hopper-Based Fully Degenerate Poly-Bernoulli Polynomials with a q-Parameter

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 121 ◽  
Author(s):  
Ugur Duran ◽  
Patrick Sadjang
Keyword(s):  
Inversion Formula ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Addition Formula ◽  
Summation Formulas

We firstly consider the fully degenerate Gould–Hopper polynomials with a q parameter and investigate some of their properties including difference rule, inversion formula and addition formula. We then introduce the Gould–Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provide some of their diverse basic identities and properties including not only addition property, but also difference rule properties. By the same way of mentioned polynomials, we define the Gould–Hopper-based fully degenerate ( α , q ) -Stirling polynomials of the second kind, and then give many relations. Moreover, we derive multifarious correlations and identities for foregoing polynomials and numbers, including recurrence relations and implicit summation formulas.

scholarly journals A generalisation of the summation formula of Plana

1999 ◽  
Vol 59 (2) ◽  
pp. 315-322
Author(s):  
Clément Frappier
Keyword(s):  
Bessel Function ◽  
Summation Formula

An extension of the classical summation formula of Plana is obtained. The extension is obtained by using the zeros of a Bessel function of the first kind.

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