In recent years, many researchers (see, for example, Araci et al. in Springer Plus5(1), Article ID 860.
https://doi.org/10.1186/s40064-016-2357-4
,
2016
to Zhang and Yang in Comput Math Appl ...56:2993–2999,
2008
) worked on the Apostol–Bernoulli type polynomials and numbers. They introduced and investigated some properties of these types of polynomials and numbers including several identities and symmetric relations for them. Carlitz (Script Math 25:323–330,
1961
, Utilitas Math 15:51–88,
1979
) introduced the degenerate Bernoulli numbers. Dolgy et al. (Adv Stud Contemp Math 26:203–209,
2016
) and Kwon et al. (Filomat 26:1–9,
2016
) introduced and investigated the modified degenerate Bernoulli polynomials and the modified degenerate Euler polynomials, respectively. They gave some relations for these polynomials. Özarslan (Comput Math Appl 62:2452–2462,
2011
) and Khan et al. (J Math Anal Appl 351:756–764,
2009
) considered the Hermite-based unified Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Khan et al. (J Nonlinear Sci Appl 10:5072–5081,
2017
) introduced the partially degenerate Hermite–Genocchi polynomials. In this article, we define the modified degenerate Hermite-based Apostol–Bernoulli, the modified degenerate Hermite-based Apostol–Euler and the modified Hermite-based Apostol–Genocchi polynomials. We prove two theorems and several symmetry relations for each of these families of polynomials. We also derive finite summation formulas for the modified degenerate unified Hermite-based Apostol type polynomials.
A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Luo and Srivastava Q.-M. Luo, H.M. Srivastava, Some generalizations of the ...Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728. In this paper, we prove several symmetry identities for these generalized Apostol type polynomials by using their generating functions. As special cases and consequences of our results, we obtain the corresponding symmetry identities for the Apostol–Euler polynomials of higher order, the Apostol–Bernoulli polynomials of higher order and the Apostol–Genocchi polynomials of higher order, and also for another family of generalized Apostol type polynomials which were investigated systematically by Ozden et al. H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010) 2779–2787. We also derive several relations between the Apostol type polynomials, the generalized sum of integer powers and the generalized alternating sum. It is shown how each of these results would extend the corresponding known identities.
In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the ...generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.
In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function
Φ
n
(
z,
s,
a) which was introduced recently by Choi et al. J. Choi, D.S. Jang, ...H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008). We then introduce and investigate some
q-extensions of the multiple Hurwitz–Lerch Zeta function
Φ
n
(
z,
s,
a), the Apostol–Bernoulli polynomials
B
k
(
n
)
(
x
;
λ
)
of order
n, and the Apostol–Euler polynomials
E
k
(
n
)
(
x
;
λ
)
of order
n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.
The Four Horsemen narrates the history of revolution in Spain, Naples, Greece, and Russia in the 1820s, connecting the social movements and activities on the ground, in the inimitable voice of a ...renowned historian.
Fourier expansions of higher-order Apostol–Genocchi and Apostol–Bernoulli polynomials are obtained using Laurent series and residues. The Fourier expansion of higher-order Apostol–Euler polynomials ...is obtained as a consequence.
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