Some properties of the generalized Apostol-type polynomials

  • Da-Qian Lu1 and

    Affiliated with

    • Qiu-Ming Luo2Email author

      Affiliated with

      Boundary Value Problems20132013:64

      DOI: 10.1186/1687-2770-2013-64

      Received: 10 December 2012

      Accepted: 19 March 2013

      Published: 28 March 2013

      Abstract

      In this paper, we study some properties of the generalized Apostol-type polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:5702-5728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials, and Apostol-Genocchi polynomials of high order.

      MSC:11B68, 33C65.

      Keywords

      generalized Apostol type polynomials recurrence relations differential equations connected problems quasi-monomial

      1 Introduction, definitions and motivation

      The classical Bernoulli polynomials B n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq1_HTML.gif, the classical Euler polynomials E n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq2_HTML.gif and the classical Genocchi polynomials G n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq3_HTML.gif, together with their familiar generalizations B n ( α ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq4_HTML.gif, E n ( α ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq5_HTML.gif and G n ( α ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq6_HTML.gif of (real or complex) order α, are usually defined by means of the following generating functions (see, for details, [1], pp.532-533 and [2], p.61 et seq.; see also [3] and the references cited therein):
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ1_HTML.gif
      (1.1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ2_HTML.gif
      (1.2)
      and
      ( 2 z e z + 1 ) α e x z = n = 0 G n ( α ) ( x ) z n n ! ( | z | < π ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ3_HTML.gif
      (1.3)
      So that, obviously, the classical Bernoulli polynomials B n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq1_HTML.gif, the classical Euler polynomials E n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq2_HTML.gif and the classical Genocchi polynomials G n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq3_HTML.gif are given, respectively, by
      B n ( x ) : = B n ( 1 ) ( x ) , E n ( x ) : = E n ( 1 ) ( x ) and G n ( x ) : = G n ( 1 ) ( x ) ( n N 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ4_HTML.gif
      (1.4)
      For the classical Bernoulli numbers B n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq7_HTML.gif, the classical Euler numbers E n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq8_HTML.gif and the classical Genocchi numbers G n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq9_HTML.gif of order n, we have
      B n : = B n ( 0 ) = B n ( 1 ) ( 0 ) , E n : = E n ( 0 ) = E n ( 1 ) ( 0 ) and G n : = G n ( 0 ) = G n ( 1 ) ( 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ5_HTML.gif
      (1.5)

      respectively.

      Some interesting analogues of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [4], p.165, Eq. (3.1)) and (more recently) by Srivastava (see [5], pp.83-84). We begin by recalling here Apostol’s definitions as follows.

      Definition 1.1 (Apostol [4]; see also Srivastava [5])

      The Apostol-Bernoulli polynomials B n ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq10_HTML.gif ( λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif) are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ6_HTML.gif
      (1.6)
      with, of course,
      B n ( x ) = B n ( x ; 1 ) and B n ( λ ) : = B n ( 0 ; λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ7_HTML.gif
      (1.7)

      where B n ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq12_HTML.gif denotes the so-called Apostol-Bernoulli numbers.

      Recently, Luo and Srivastava [6] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of order α.

      Definition 1.2 (Luo and Srivastava [6])

      The Apostol-Bernoulli polynomials B n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq13_HTML.gif ( λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif) of order α N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq14_HTML.gif are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ8_HTML.gif
      (1.8)
      with, of course,
      B n ( α ) ( x ) = B n ( α ) ( x ; 1 ) and B n ( α ) ( λ ) : = B n ( α ) ( 0 ; λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ9_HTML.gif
      (1.9)

      where B n ( α ) ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq15_HTML.gif denotes the so-called Apostol-Bernoulli numbers of order α.

      On the other hand, Luo [7], gave an analogous extension of the generalized Euler polynomials as the so-called Apostol-Euler polynomials of order α.

      Definition 1.3 (Luo [7])

      The Apostol-Euler polynomials E n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq16_HTML.gif ( λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif) of order α N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq14_HTML.gif are defined by means of the following generating function:
      ( 2 λ e z + 1 ) α e x z = n = 0 E n ( α ) ( x ; λ ) z n n ! ( | z | < | log ( λ ) | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ10_HTML.gif
      (1.10)
      with, of course,
      E n ( α ) ( x ) = E n ( α ) ( x ; 1 ) and E n ( α ) ( λ ) : = E n ( α ) ( 0 ; λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ11_HTML.gif
      (1.11)

      where E n ( α ) ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq17_HTML.gif denotes the so-called Apostol-Euler numbers of order α.

      On the subject of the Genocchi polynomials G n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq3_HTML.gif and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, [814]). Moreover, Luo (see [1214]) introduced and investigated the Apostol-Genocchi polynomials of (real or complex) order α, which are defined as follows:

      Definition 1.4 The Apostol-Genocchi polynomials G n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq18_HTML.gif ( λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif) of order α N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq14_HTML.gif are defined by means of the following generating function:
      ( 2 z λ e z + 1 ) α e x z = n = 0 G n ( α ) ( x ; λ ) z n n ! ( | z | < | log ( λ ) | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ12_HTML.gif
      (1.12)
      with, of course,
      G n ( α ) ( x ) = G n ( α ) ( x ; 1 ) , G n ( α ) ( λ ) : = G n ( α ) ( 0 ; λ ) , G n ( x ; λ ) : = G n ( 1 ) ( x ; λ ) and G n ( λ ) : = G n ( 1 ) ( λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ13_HTML.gif
      (1.13)

      where G n ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq19_HTML.gif, G n ( α ) ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq20_HTML.gif and G n ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq21_HTML.gif denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of order α and the Apostol-Genocchi polynomials, respectively.

      Recently, Luo and Srivastava [15] introduced a unification (and generalization) of the above-mentioned three families of the generalized Apostol type polynomials.

      Definition 1.5 (Luo and Srivastava [15])

      The generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq22_HTML.gif ( α N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq14_HTML.gif, λ , u , v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq23_HTML.gif) of order α are defined by means of the following generating function:
      ( 2 u z v λ e z + 1 ) α e x z = n = 0 F n ( α ) ( x ; λ ; u , v ) z n n ! ( | z | < | log ( λ ) | ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ14_HTML.gif
      (1.14)
      where
      F n ( α ) ( λ ; u , v ) : = F n ( α ) ( 0 ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ15_HTML.gif
      (1.15)

      denote the so-called Apostol type numbers of order α.

      So that, by comparing Definition 1.5 with Definitions 1.2, 1.3 and 1.4, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ16_HTML.gif
      (1.16)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ17_HTML.gif
      (1.17)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ18_HTML.gif
      (1.18)
      A polynomial p n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq24_HTML.gif ( n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq25_HTML.gif, x C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq26_HTML.gif) is said to be a quasi-monomial [16], whenever two operators M ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq27_HTML.gif, P ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq28_HTML.gif, called multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ19_HTML.gif
      (1.19)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ20_HTML.gif
      (1.20)
      which can be combined to get the identity
      M ˆ P ˆ p n ( x ) = n p n ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ21_HTML.gif
      (1.21)
      The Appell polynomials [17] can be defined by considering the following generating function:
      A ( t ) e x t = n = 0 R n ( x ) n ! t n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ22_HTML.gif
      (1.22)
      where
      A ( t ) = k = 0 R k k ! t k ( A ( 0 ) 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ23_HTML.gif
      (1.23)

      is analytic function at t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq29_HTML.gif.

      From [18], we know that the multiplicative and derivative operators of R n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq30_HTML.gif are
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ24_HTML.gif
      (1.24)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ25_HTML.gif
      (1.25)
      where
      A ( t ) A ( t ) = n = 0 α n t n n ! . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ26_HTML.gif
      (1.26)

      By using (1.21), we have the following lemma.

      Lemma 1.6 ([18])

      The Appell polynomials R n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq30_HTML.gif defined by (1.22) satisfy the differential equation:
      α n 1 ( n 1 ) ! y ( n ) + α n 2 ( n 2 ) ! y ( n 1 ) + + α 1 1 ! y + ( x + α 0 ) y n y = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ27_HTML.gif
      (1.27)
      where the numerical coefficients α k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq31_HTML.gif, k = 1 , 2 , , n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq32_HTML.gif are defined in (1.26), and are linked to the values R k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq33_HTML.gif by the following relations:
      R k + 1 = h = 0 k ( k h ) R h α k h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equa_HTML.gif
      Let P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq34_HTML.gif be the vector space of polynomials with coefficients in ℂ. A polynomial sequence { P n } n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq35_HTML.gif be a polynomial set. { P n } n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq35_HTML.gif is called a σ-Appell polynomial set of transfer power series A is generated by
      G ( x , t ) = A ( t ) G 0 ( x , t ) = n = 0 P n ( x ) n ! t n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ28_HTML.gif
      (1.28)
      where G 0 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq36_HTML.gif is a solution of the system:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equb_HTML.gif

      In [19], the authors investigated the connection coefficients between two polynomials. And there is a result about connection coefficients between two σ-Appell polynomial sets.

      Lemma 1.7 ([19])

      Let σ Λ ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq37_HTML.gif. Let { P n } n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq38_HTML.gif and { Q n } n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq39_HTML.gif be two σ-Appell polynomial sets of transfer power series, respectively, A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq40_HTML.gif and A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq41_HTML.gif. Then
      Q n ( x ) = m = 0 n n ! m ! α n m P m ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ29_HTML.gif
      (1.29)
      where
      A 2 ( t ) A 1 ( t ) = k = 0 α k t k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equc_HTML.gif

      In recent years, several authors obtained many interesting results involving the related Bernoulli polynomials and Euler polynomials [5, 2040]. And in [29], the authors studied some series identities involving the generalized Apostol type and related polynomials.

      In this paper, we study some other properties of the generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq42_HTML.gif, including the recurrence relations, the differential equations and some connection problems, which extend some known results. As special, we obtain some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials and Apostol-Genocchi polynomials of high order.

      2 Recursion formulas and differential equations

      From the generating function (1.14), we have
      x F n ( α ) ( x ; λ ; u , v ) = n F n 1 ( α ) ( x ; λ ; u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ30_HTML.gif
      (2.1)

      A recurrence relation for the generalized Apostol type polynomials is given by the following theorem.

      Theorem 2.1 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq22_HTML.gif holds true:
      ( α v n + 1 1 ) F n + 1 ( α ) ( x ; λ ; u , v ) = α λ 2 u n ! ( n + v ) ! F n + v ( α + 1 ) ( x + 1 ; λ ; u , v ) x F n ( α ) ( x ; λ ; u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ31_HTML.gif
      (2.2)

      Proof Differentiating both sides of (1.14) with respect to t, and using some elementary algebra and the identity principle of power series, recursion (2.2) easily follows. □

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 2.1, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have:

      Corollary 2.2 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol-Bernoulli polynomials B n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq49_HTML.gif holds true:
      [ α ( n + 1 ) ] B n + 1 ( α ) ( x ; λ ) = α λ B n + 1 ( α + 1 ) ( x + 1 ; λ ) x B n ( α ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ32_HTML.gif
      (2.3)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq51_HTML.gif in Theorem 2.1, we have the following corollary.

      Corollary 2.3 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol-Euler polynomials E n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq16_HTML.gif holds true:
      E n + 1 ( α ) ( x ; λ ) = x E n ( α ) ( x ; λ ) α λ 2 E n ( α + 1 ) ( x + 1 ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ33_HTML.gif
      (2.4)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 2.1, we have the following corollary.

      Corollary 2.4 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol-Genocchi polynomials G n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq18_HTML.gif holds true:
      2 [ α ( n + 1 ) ] G n + 1 ( α ) ( x ; λ ) = α λ G n + 1 ( α + 1 ) ( x + 1 ; λ ) 2 ( n + 1 ) x G n ( α ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ34_HTML.gif
      (2.5)
      From (1.14) and (1.22), we know that the generalized Appostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq42_HTML.gif is Appell polynomials with
      A ( t ) = ( 2 u t v λ e t + 1 ) α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ35_HTML.gif
      (2.6)
      From the Eq. (23) of [15], we know that G 0 ( 1 ; λ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq52_HTML.gif. So from (2.6) and (1.12), we can obtain that if v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq51_HTML.gif, we have
      A ( t ) A ( t ) = λ α 2 n = 0 G n + 1 ( 1 ; λ ) n + 1 t n n ! . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ36_HTML.gif
      (2.7)
      By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators of the generalized Appostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq22_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ37_HTML.gif
      (2.8)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ38_HTML.gif
      (2.9)
      From (2.1), we can obtain
      p x p F n ( α ) ( x ; λ ; u , v ) = n ! ( n p ) ! F n p ( α ) ( x ; λ ; u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ39_HTML.gif
      (2.10)

      Then by using (1.20), (2.8) and (2.10), we obtain the following result.

      Theorem 2.5 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq53_HTML.gif holds true:
      F n + 1 ( α ) ( x ; λ ; u , 0 ) = ( x + λ α 2 G 1 ( 1 ; λ ) ) F n ( α ) ( x ; λ ; u , 0 ) + λ α 2 k = 0 n 1 ( n k ) G n k + 1 ( 1 ; λ ) n k + 1 F n k ( α ) ( x ; λ ; u , 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ40_HTML.gif
      (2.11)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif in Theorem 2.5, we have the following corollary.

      Corollary 2.6 For any integral n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq43_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq11_HTML.gif and α N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq44_HTML.gif, the following recurrence relation for the generalized Apostol-Euler polynomials E n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq54_HTML.gif holds true:
      E n + 1 ( α ) ( x ; λ ) = ( x + λ α 2 G 1 ( 1 ; λ ) ) E n ( α ) ( x ; λ ) + λ α 2 k = 0 n 1 ( n k ) G n k + 1 ( 1 ; λ ) n k + 1 E n k ( α ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ41_HTML.gif
      (2.12)

      Furthermore, applying Lemma 1.7 to F n ( α ) ( x ; λ ; u , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq55_HTML.gif, we have the following theorem.

      Theorem 2.7 The generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq55_HTML.gif satisfy the differential equation:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ42_HTML.gif
      (2.13)

      Specially, by setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif in Theorem 2.7, then we have the following corollary.

      Corollary 2.8 The generalized Apostol-Euler polynomials E n ( α ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq54_HTML.gif satisfy the differential equation:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ43_HTML.gif
      (2.14)

      3 Connection problems

      From (1.14) and (1.28), we know that the generalized Apostol type polynomials F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq56_HTML.gif are a D x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq57_HTML.gif-Appell polynomial set, where D x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq57_HTML.gif denotes the derivative operator.

      From Table 1 in [19], we know that the derivative operators of monomials x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq58_HTML.gif and the Gould-Hopper polynomials g n m ( x , h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq59_HTML.gif [30] are all D x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq57_HTML.gif. And their transfer power series A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq60_HTML.gif are 1 and e h t m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq61_HTML.gif, respectively.

      Applying Lemma 1.7 to P n ( x ) = x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq62_HTML.gif and Q n ( x ) = F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq63_HTML.gif, we have the following theorem.

      Theorem 3.1
      F n ( α ) ( x ; λ ; u , v ) = m = 0 n ( n m ) F n m ( α ) ( λ ; u , v ) x m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ44_HTML.gif
      (3.1)

      where F n ( α ) ( λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq64_HTML.gif is the so-called Apostol type numbers of order α defined by (1.15).

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.1, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have the following corollary.

      Corollary 3.2
      B n ( α ) ( x ; λ ) = m = 0 n ( n m ) B n m ( α ) ( λ ) x m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ45_HTML.gif
      (3.2)

      which is just Eq. (3.1) of [23].

      By setting u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq51_HTML.gif in Theorem 3.1, we have the following corollary.

      Corollary 3.3
      E n ( α ) ( x ; λ ) = m = 0 n ( n m ) E n m ( α ) ( λ ) x m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ46_HTML.gif
      (3.3)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.1, we have the following corollary.

      Corollary 3.4
      G n ( α ) ( x ; λ ) = m = 0 n ( n m ) G n m ( α ) ( λ ) x m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ47_HTML.gif
      (3.4)

      which is just Eq. (24) of [15].

      Applying Lemma 1.7 to P n ( x ) = F n ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq65_HTML.gif and Q n ( x ) = F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq63_HTML.gif, we have the following theorem.

      Theorem 3.5
      F n ( α ) ( x ; λ ; u , v ) = m = 0 n ( n m ) F n m ( α 1 ) ( λ ; u , v ) F m ( x ; λ ; u , v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ48_HTML.gif
      (3.5)

      where F n ( α ) ( λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq64_HTML.gif is the so-called Apostol type numbers of order α defined by (1.15).

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.5, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have the following corollary.

      Corollary 3.6
      B n ( α ) ( x ; λ ) = m = 0 n ( n m ) B n m ( α 1 ) ( λ ) B m ( x ; λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ49_HTML.gif
      (3.6)

      which is just Eq. (3.2) of [23].

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq51_HTML.gif in Theorem 3.5, we have the following corollary.

      Corollary 3.7
      E n ( α ) ( x ; λ ) = m = 0 n ( n m ) E n m ( α 1 ) ( λ ) E m ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ50_HTML.gif
      (3.7)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.5, we have the following corollary.

      Corollary 3.8
      G n ( α ) ( x ; λ ) = m = 0 n ( n m ) G n m ( α 1 ) ( λ ) G m ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ51_HTML.gif
      (3.8)

      Applying Lemma 1.7 to P n ( x ) = g n m ( x , h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq66_HTML.gif and Q n ( x ) = F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq63_HTML.gif, we have the following theorem.

      Theorem 3.9
      F n ( α ) ( x ; λ ; u , v ) = r = 0 n n ! r ! [ k = 0 [ ( n r ) / m ] ( 1 ) k h k k ! ( n r m k ) ! F n r m k ( α ) ( λ ; u , v ) ] g r m ( x , h ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ52_HTML.gif
      (3.9)

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.9, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have the following corollary.

      Corollary 3.10
      B n ( α ) ( x ; λ ) = r = 0 n n ! r ! [ k = 0 [ ( n r ) / m ] ( 1 ) k h k k ! ( n r m k ) ! B n r m k ( α ) ( λ ) ] g r m ( x , h ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ53_HTML.gif
      (3.10)

      which is just Eq. (3.3) of [23].

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq51_HTML.gif in Theorem 3.9, we have the following corollary.

      Corollary 3.11
      E n ( α ) ( x ; λ ) = r = 0 n n ! r ! [ k = 0 [ ( n r ) / m ] ( 1 ) k h k k ! ( n r m k ) ! E n r m k ( α ) ( λ ) ] g r m ( x , h ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ54_HTML.gif
      (3.11)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.9, we have the following corollary.

      Corollary 3.12
      G n ( α ) ( x ; λ ) = r = 0 n n ! r ! [ k = 0 [ ( n r ) / m ] ( 1 ) k h k k ! ( n r m k ) ! G n r m k ( α ) ( λ ) ] g r m ( x , h ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ55_HTML.gif
      (3.12)

      When v α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq67_HTML.gif, applying Lemma 1.7 to P n ( x ) = E n ( α 1 ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq68_HTML.gif and Q n ( x ) = F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq63_HTML.gif, we have the following theorem.

      Theorem 3.13 If v α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq69_HTML.gif, then we have
      F n ( α ) ( x ; λ ; u , v ) = m = 0 n ( n m ) 2 ( u 1 ) α G n m ( λ ) E m ( α 1 ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ56_HTML.gif
      (3.13)

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.13, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have the following corollary.

      Corollary 3.14
      B n ( x ; λ ) = 1 2 m = 0 n ( n m ) G n m ( λ ) x m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ57_HTML.gif
      (3.14)

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.13, we have the following corollary.

      Corollary 3.15
      G n ( x ; λ ) = 1 2 m = 0 n ( n m ) G n m ( λ ) x m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ58_HTML.gif
      (3.15)

      which is just the case of α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq70_HTML.gif in (3.4).

      When v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif or α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq71_HTML.gif, applying Lemma 1.7 to P n ( x ) = G n ( α 1 ) ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq72_HTML.gif and Q n ( x ) = F n ( α ) ( x ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq73_HTML.gif, we can obtain the following theorem.

      Theorem 3.16 If v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif or α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq71_HTML.gif, we have
      F n ( α ) ( x ; λ ; u , v ) = m = 0 n ( n m ) 2 ( u 1 ) α G n m ( λ ) G m ( α 1 ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ59_HTML.gif
      (3.16)

      By setting λ : = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq45_HTML.gif, u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq46_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.13, and then multiplying ( 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq48_HTML.gif on both sides of the result, we have the following corollary.

      Corollary 3.17
      B n ( α ) ( x ; λ ) = m = 0 n ( n m ) ( 1 2 ) α G n m ( λ ) G m ( α 1 ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ60_HTML.gif
      (3.17)

      When α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq70_HTML.gif in (3.17), it is just (3.15).

      By setting u = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq50_HTML.gif and v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq47_HTML.gif in Theorem 3.16, we have the following corollary.

      Corollary 3.18
      G n ( α ) ( x ; λ ) = m = 0 n ( n m ) G n m ( λ ) G m ( α 1 ) ( x ; λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ61_HTML.gif
      (3.18)

      which is equal to (3.8).

      If α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq71_HTML.gif in Theorem 3.16, we have:

      Corollary 3.19
      x n = m = 0 n ( n m ) G n m ( λ ) G m ( 1 ) ( x ; λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ62_HTML.gif
      (3.19)

      4 Hermite-based generalized Apostol type polynomials

      Finally, we give a generation of the generalized Apostol type polynomials.

      The two-variable Hermite-Kampé de Fériet polynomials (2VHKdFP) H n ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq74_HTML.gif are defined by the series [31]
      H n ( x , y ) = n ! r = 0 [ n / 2 ] x n 2 r y r r ! ( n 2 r ) ! http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ63_HTML.gif
      (4.1)
      with the following generating function:
      exp ( x t + y t 2 ) = n = 0 t n n ! H n ( x , y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ64_HTML.gif
      (4.2)
      And the 2VHKdFP H n ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq74_HTML.gif are also defined through the operational identity
      exp ( y 2 x 2 ) { x n } = H n ( x , y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ65_HTML.gif
      (4.3)
      Acting the operator exp ( y 2 x 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq75_HTML.gif on (1.14), and by the identity [32]
      exp ( y 2 x 2 ) { exp ( a x 2 + b x ) } = 1 1 + 4 a y exp ( a x 2 b x b 2 y 1 + 4 a y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ66_HTML.gif
      (4.4)
      we define the Hermite-based generalized Apostol type polynomials F n ( α ) H ( x , y ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq76_HTML.gif by the generating function
      ( 2 u z v λ e t + 1 ) α e x t + y t 2 = n = 0 H F n ( α ) ( x , y ; λ ; u , v ) t n n ! ( | t | < | log ( λ ) | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ67_HTML.gif
      (4.5)
      Clearly, we have
      F n H ( x , y ; λ ; u , v ) = H F n ( 1 ) ( x , y ; λ ; u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equd_HTML.gif
      From the generating function (4.5), we easily obtain
      x H F n ( α ) ( x , y ; λ ; u , v ) = n H F n 1 ( α ) ( x , y ; λ ; u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ68_HTML.gif
      (4.6)
      and
      y H F n ( α ) ( x , y ; λ ; u , v ) = n ( n 1 ) H F n 2 ( α ) ( x , y ; λ ; u , v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ69_HTML.gif
      (4.7)
      which can be combined to get the identity
      2 x 2 H F n ( α ) ( x , y ; λ ; u , v ) = y H F n ( α ) ( x , y ; λ ; u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ70_HTML.gif
      (4.8)
      Acting with the operator exp y 2 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq77_HTML.gif on both sides of (3.1), (3.5), (3.13), (3.18), and by using (4.3), we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ71_HTML.gif
      (4.9)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ72_HTML.gif
      (4.10)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ73_HTML.gif
      (4.11)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Equ74_HTML.gif
      (4.12)
      where E n ( α ) H ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq78_HTML.gif and G n ( α ) H ( x ; λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_IEq79_HTML.gif are the Hermite-based generalized Apostol-Euler polynomials and the Hermite-based generalized Apostol-Genocchi polynomials respectively, defined by the following generating functions:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-64/MediaObjects/13661_2012_Article_321_Eque_HTML.gif

      Declarations

      Acknowledgements

      Dedicated to Professor Hari M Srivastava.

      The present investigation was supported, in part, by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625 and Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Yangzhou University
      (2)
      Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus

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