Some New Classes of Generalized Apostol-Euler and Apostol-Genocchi Polynomials

Table 1

𝑥^𝑛 expressed in terms of sums of special polynomials or numbers.

No.	Special polynomials or numbers	Series representation for 𝑥^𝑛
(1)	Hermite polynomials [27, page 194, (4)]	𝑥^𝑛=𝑛!2^𝑛[𝑛/2]𝑘∑=0𝐻𝑛−2𝑘(𝑥)𝑘!(𝑛−2𝑘)!
(2)	Legendre polynomials [27, page 181, Theorem 65]	𝑥^𝑛=𝑛!2^𝑛[𝑛/2]𝑘∑=0(2𝑛−4𝑘+1)𝑃𝑛−2𝑘(𝑥)𝑘!(3/2)𝑛−𝑘
(3)	Generalized Laguerre polynomials [27, page 207, (2)]	𝑥^𝑛=𝑛!(1+𝛼)∑𝑛𝑛𝑘=0(−1)^𝑘𝐿^𝑘(𝛼)(𝑥)(1+𝛼)𝑘(𝑛−𝑘)!
(4)	Gegenbauer polynomials [27, page 283, (36)]	𝑥^𝑛=𝑛!2^𝑛[𝑛/2]𝑘∑=0(𝜈+𝑛−2𝑘)𝐶^𝜈𝑛−2𝑘(𝑥)𝑘!(𝜈)𝑛+1−𝑘
(5)	Stirling numbers of the second kind [5, page 207, Theorem B]	𝑥^𝑛=∑^𝑛𝑘=0⎛⎜⎜⎝𝑥𝑘⎞⎟⎟⎠𝑘!𝑆(𝑛,𝑘)
(6)	Bernoulli polynomials [7, page 26]	𝑥^𝑛=1𝑛+1^𝑛𝑘∑=0⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠𝐵𝑘(𝑥)
(7)	Euler polynomials [7, page 30]	𝑥^𝑛=12⎡⎢⎢⎣𝐸^𝑛(𝑥)+∑^𝑛𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐸𝑘(𝑥)⎤⎥⎥⎦
(8)	Apostol-Bernoulli polynomials [8, page 634, (29)]	𝑥^𝑛=1𝑛+1⎡⎢⎢⎣𝜆^𝑛+1𝑘∑=0⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠𝔅𝑘(𝑥;𝜆)−𝔅𝑛+1(𝑥;𝜆)⎤⎥⎥⎦
(9)	Apostol-Euler polynomials [8, page 635, (32)]	𝑥^𝑛=12⎡⎢⎢⎣𝜆^𝑛𝑘∑=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝔈𝑘(𝑥;𝜆)+𝔈𝑛(𝑥;𝜆)⎤⎥⎥⎦
(10)	Generalized Apostol-Euler polynomials [28, page 1325, (2.4)]	𝑥^𝑛=12^𝛽∞𝑘∑=0⎛⎜⎜⎝𝛽𝑘⎞⎟⎟⎠𝜆𝑘𝔈^𝑛(𝛽)(𝑥+𝑘;𝜆),𝛽∈ℂ
(11)	Generalized Bernoulli polynomials and Stirling numbers [28, page 1329, (2.16)]	𝑥^𝑛=∑^𝑛𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠⎛⎜⎜⎝𝑙+𝑗𝑗⎞⎟⎟⎠^−1𝑆(𝑙+𝑗,𝑗)𝐵^(𝑗)𝑛−𝑙(𝑥),𝑗∈ℕ0
(12)	Generalized Apostol-Bernoulli polynomials and generalized Stirling numbers [28, page 1329, (2.15)]	𝑥^𝑛=𝑛!∑^𝑛𝑙=−𝑗𝑗!(𝑙+𝑗)!(𝑛−𝑙)!𝑆(𝑙+𝑗,𝑗;𝜆)𝔅^(𝑗)𝑛−𝑙(𝑥;𝜆),𝑗∈ℕ0
(13)	Generalized Bernoulli polynomials [29, page 158, (2.6)]	𝑥^𝑛=∑^𝑛𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑘!(𝑘+𝑚)!𝐵^[𝑚−1]𝑛−𝑘(𝑥),𝑚∈ℕ