On certain generalized q-appell polynomial expansions

Transcription

1 doi: /umcsmath ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL LXVIII, NO 2, 2014 SECTIO A THOMAS ERNST On certain generalized -Appell polynomial expansions Abstract We study -analogues of three Appell polynomials, the H-polynomials, the Apostol Bernoulli and Apostol Euler polynomials, whereby two new -difference operators and the NOVA -addition play ey roles The definitions of the new polynomials are by the generating function; lie in our boo, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials In order to find a certain formula, we introduce a -logarithm We conclude with a brief discussion of multiple -Appell polynomials 1 Introduction The aim of this paper is to describe how the -umbral calculus extends in a natural way to produce -analogues of conversion theorems and polynomial expansions for the following Appell polynomials: H- polynomials, Apostol Bernoulli and Apostol Euler from the recent articles on this theme, as well as multiple variable extensions To this aim, we use certain -difference operators nown from the boo [3], and some new operators containing a factor λ from the previous wor of Luo and Srivastava [8], [9], [10] on Apostol Bernoulli polynomials The -Appell polynomials have been used before in [4], where their basic definition was given together with 2010 Mathematics Subject Classification Primary 05A40; 11B68; Secondary 05A10 Key words and phrases -Apostol Bernoulli polynomials, -Apostol Euler polynomials, -H-polynomials, complementary argument theorem, generating function, multiple -Appell polynomials

2 28 T Ernst several matrix applications The umbral method [3], influenced by Jordan [5] and Nørlund [12], forms the basis for the terminology and umbral method, which enables convenient -analogues of the formulas for Appell polynomials; our formulas resemble the Appell polynomial formulas in a remarable way A certain -Taylor formula plays a ey role in many proofs This paper is organized as follows: In this section we give the general definitions In each section, we then give the specific definitions and special values which we use there In Section 2, we introduce two dual polynomials together with recursion formulas, symmetry relations and complementary argument theorem Let λ R and let E (x denote the -exponential function In Sections 3 and 4 in the spirit of Apostol, Luo and Srivastava, we introduce and discuss two dual forms of the generalized -Apostol Bernoulli polynomials, together with the many applications that were mentioned earlier In Sections 5 and 6, we continue the discussion with two dual forms of the generalized -Apostol Euler polynomials Two of their generating functions are given below: (1 t n (λe (t 1 n E (xt t ν NWA,λ,ν, (x, {ν}! and (2 2 n (λe (t+1 n E (xt t ν F (n NWA,λ,ν, (x {ν}! This is followed by formulas which contain both inds of these polynomials Many of the formulas are proved by simple manipulations of the generating functions In Section 7 we show that the many expansion formulas according to Nørlund can also be formulated for our polynomials In Section 7 we extend the previous considerations to a more general form, named multiplicative -Appell polynomials More on this will come in a future paper We now come to Section 9; in order to find -analogues of the corresponding formulas for the generating functions, we, formally, introduce a logarithm for the -exponential function The calculations are valid for so-called -real numbers In Section 10, we briefly discuss multiple -Appell polynomials We now start with the definitions, compare with the boo [3] Some of the notation is well nown and will be sipped Definition 1 Let the Gauss -binomial coefficient be defined by (3 ( n {n}!, 0, 1,,n {}!{n }!

3 On certain generalized -Appell polynomial expansions 29 Let a and b be any elements with commutative multiplication Then the NWA -addition is given by n ( n (4 (a b n a b n, n 0, 1, 2, The JHC -addition is the function n ( ( n (5 (a b n ( 2 b a n a n b a ;, n 0, 1, 2, If 0 < < 1 and z < 1 1, the -exponential function is defined by 1 (6 E (z {}! z The -derivative is defined by f(x f(x (1 x, if C\{1}, x 0; (7 (D f(x df dx (x if 1; (0 if x 0 df dx Definition 2 Let the NWA -shift operator be given by (8 E( (x n (x 1 n Definition 3 The related JHC -shift operator is given by (9 E( (x n (x 1 n Let I denote the identity operator The Apostol NWA -difference operator is given by (10 NWA,A, λe( I The Apostol NWA -mean value operator is given by (11 NWA,A, λe( +I 2 The Apostol JHC -difference operator is given by (12 JHC,A, λe( I The Apostol JHC -mean value operator is given by (13 JHC,A, λe( +I 2 Definition 4 For every power series f n (t, the -Appell polynomials or Φ polynomials of degree ν and order n have the following generating function: t ν (14 f n (te (xt {ν}! Φ(n ν, (x n

4 30 T Ernst For x 0we get the Φ (n ν, number of degree ν and order n Definition 5 For f n (t of the form h(t n, we call the -Appell polynomial Φ in (14 multiplicative Theorem 11 We have the two -Taylor formulas (15 Φ (n ν, (x y Φ (n ν, (xy (16 Φ (n ν, (x y ( ν ( (n 2 Φ ν, (xy 2 The H polynomials First, we repeat some of the definitions of certain related -Appell polynomials for later use These polynomials are more general forms of the polynomials we wish to study Definition 6 The generating function for β ν, (n (x is t n g(t (17 (E (t 1 n E t ν β ν, (n (x (xt {ν}! Definition 7 The generating function for γ ν, (n (x is given by t n g(t (18 (t 1 n E t ν γ ν, (n (x (xt {ν}! (E 1 Definition 8 The generating function for η ν, (n (x is given by 2 n (19 (E (t+1 n g(te t ν η ν, (n (x (xt {ν}! Definition 9 The generating function for θ ν, (n (x is given by 2 n (20 (t+1 n g(te t ν θ ν, (n (x (xt {ν}! (E 1 We now come to the generating function of the polynomials we want to study in this section (for 1: The H polynomials are defined in [14, p 532 (37] (21 2t e t +1 ext t ν H ν (x ν! The odd H numbers are zero and the even H numbers are expressable in terms of Bernoulli numbers as (22 H 2n 2(1 2 2n B 2n

5 On certain generalized -Appell polynomial expansions 31 Definition 10 The generating function for H (n NWA,ν, (x is a special case of (19: (23 (2t n (E (t+1 n E (xt t ν H (n NWA,ν, (x, t < 2π {ν}! Definition 11 The generating function for H (n JHC,ν, (x is a special case of (20: (24 (E 1 (2t n (t+1 n E (xt t ν H (n JHC,ν,, (x, t < 2π {ν}! The polynomials in (23 and (24 are -analogues of the generalized H polynomials We now turn to these -analogues Theorem 21 We have (25 NWA, H (n NWA,ν, (x {ν} H (n 1 NWA,ν 1, (x, (26 H NWA,0, 0, H NWA,1, 1, (H NWA, 1 + H NWA,, 0, >1 (27 JHC, H (n JHC,ν,, (x {ν} H (n 1 JHC,ν, 1, (x, (28 H JHC,0, 0, H JHC,1, 1, (H JHC, 1 + H JHC,, 0, >1 The following table lists some of the first H NWA,ν, numbers n 2 n 3 n 4 ( ( (1 + 2 (1 + 2 ( n 5 ( 1+(1 + ( {3} {5} 2 4 We need not calculate the H JHC,ν,, numbers, since we have the following symmetry relations: Theorem 22 For ν even, H NWA,ν, H JHC,ν, (29 For ν uneven, H NWA,ν, H JHC,ν,, ν > 1 For the convenience of the reader, we mae a short repetition: Definition 12 The Ward -Bernoulli numbers [17, p 265, 164], are given by (30 B NWA,n, B (1 NWA,n,

6 32 T Ernst The following table lists some of the first Ward -Bernoulli numbers n 0 n 1 n 2 n 3 1 ( ({3}! 1 (1 3 ({2} 1 ({4} 1 n 4 4 ( ({2} 2 {3} {5} 1 The following three formulas express x n in terms of -Appell polynomials Lemma 23 (31 x n 1 n ( n +1 B NWA,, (x {n +1} (see [3, 4149, p 121] [ (32 x ν 1 F NWA,ν, (x+ 2 ( ] ν F NWA,, (x (see [3, 4206, p 130] [ (33 x ν 1 ( ] ν +1 H NWA,ν+1, (x+ H NWA,+1, (x 2{ν +1} +1 Proof For the last formula, use formula (25 Theorem 24 (A -analogue of [13, p 489] Let Φ (n ν, (x be a -Appell polynomial Then the following three addition formulas apply: Φ (n ν, (x y ( ( (34 1 ν j +1 Φ (n ν j, {j +1} j j (y B NWA,, (x Φ (n ν, (x y 1 Φ (n 2 ν, (y (35 (36 Φ (n ν, (x y 1 2 F NWA,, (x+ H NWA,+1, (x+ j0 ( j F NWA,j, (x Φ (n ν, (y 1 { +1} ( +1 H NWA,j+1, (x j +1 j0

7 On certain generalized -Appell polynomial expansions 33 Proof Use formulas (31 (33 and (15 The following complementary argument theorems extend the ones given in [3, p 153] Theorem 25 A -analogue of the Raabe Bernoulli complementary argument theorem [11, p 128, (1]: (37 JHC,ν, (x ( 1ν NWA,ν, (n x Theorem 26 A -analogue of the Euler complementary argument theorem by Milne Thomson [11, p 145, (1]: (38 F (n JHC,ν, (x ( 1ν F (n NWA,ν, (n x These two formulas can be generalized to Theorem 27 (A -analogue of the Raabe Bernoulli complementary argument theorem [11, p 128, (1] Assume that g(t in (17 and (18 are eual and even functions Then (39 γ ν, (n (x ( 1 ν β ν, (n (n x Theorem 28 (A -analogue of the Euler complementary argument theorem by Milne Thomson [11, p 145, (1] Assume that g(t in (19 and (20 are eual and even functions Then (40 θ ν, (n (x ( 1 ν η ν, (n (n x Theorem 29 A special case of (40, and a -analogue of [14, p 532]: (41 H (n JHC,ν, (x ( 1ν+n H (n NWA,ν, ((n x Proof Use the generating function Corollary 210 A -analogue of a generalization of [14, p 532] (42 H (n NWA,ν, (x+( 1ν+n H (n JHC,ν, ( x 2{ν} H (n 1 NWA,ν 1, (x Proof Use the generating function and (25, (41 3 The NWA -Apostol-Bernoulli polynomials Throughout, we assume that λ 0 The b polynomials are more general forms of the NWA -Apostol Bernoulli polynomials, which we will study in this section Definition 13 The polynomials b (n λ,ν, (x are defined by (43 t n g(t (λe (t 1 n E (xt t ν b (n λ,ν, (x {ν}! The generating function for B NWA,ν, (x is a special case of (44:

8 34 T Ernst Definition 14 The generalized NWA -Apostol Bernoulli polynomials (x are defined by NWA,λ,ν, (44 t n (λe (t 1 n E (xt t ν NWA,λ,ν, (x, t +logλ < 2π {ν}! Assume that λ 1 The poles in the denominator of (44 are the roots of E (t λ 1, which implies that in some cases the limit λ 1 is not straightforward and needs some further consideration We have (45 NWA,A, b (n λ,ν, (x {ν} b (n 1 λ,ν 1, (x D b (n 1 λ,ν, (x This leads to the following recurrence for the NWA -Apostol Bernoulli numbers: (46 B NWA,λ,0, 0, λ(b NWA,λ, 1 B NWA,λ,, δ 1,, > 0 The following table lists some of the first B NWA,λ,ν, numbers ν 0 ν 1 ν 2 ν λ(1 + λ{3} 0 λ(1 1 λ (1 λ 2 (λ 1 3 ν 4 ν 5 λ{2} (1 + 2 (1 + 2λ +2λ 2 + λ 2 3 Aλ{5} (1 λ 4 (λ 1 5 where (47 A 1+3λ +4λ 2 + λ λ 3 (3 + +λ 2 2 ( Theorem 31 (A generalization of [3, 4242] If s l1 n l n, (48 NWA,λ,, (x 1 x s ( m 1,,m s m 1 ++m s s j1 B (n j NWA,λ,m j, (x j, where we assume that n j operates on x j Corollary 32 (A -analogue of [8, p 300 (49] ( (49 ν NWA,λ,ν, (x B (n 1 NWA,λ,ν, B NWA,λ,,(x Corollary 33 A -analogue of [9, p 634, (28, (29]: B (n 1 NWA,λ,ν, (x [ (50 1 {ν +1} ν+1 +1 λ NWA,λ,, (x B(n NWA,λ,ν+1, (x ]

9 On certain generalized -Appell polynomial expansions 35 A generalization of [3, 4149] [ (51 x ν 1 ν+1 ( ] ν +1 λ B NWA,λ,, (x B NWA,λ,ν+1, (x {ν +1} 4 The JHC -Apostol Bernoulli polynomials The c polynomials are more general forms of the JHC -Apostol Bernoulli polynomials, which we will study in this section Definition 15 The polynomials c (n λ,ν, (x are defined by t n g(t (52 (λe 1 (t 1 n E t ν c (n λ,ν, (xt (x {ν}! Definition 16 The generalized JHC -Apostol Bernoulli polynomials (x are defined by JHC,λ,ν, t n (53 (λe 1 (t 1 n E (xt We have t ν JHC,λ,ν, (x, t +logλ < 2π {ν}! (54 JHC,A, c (n λ,ν, (x {ν} c (n 1 λ,ν 1, (x D c (n 1 λ,ν, (x The defining relation of the JHC -Apostol Bernoulli numbers is given by the umbral recurrence (55 B JHC,λ,0, 0, λ(b JHC,λ, 1 B JHC,λ,, δ 1,, > 0 We need not calculate the B JHC,λ,ν, numbers, since we have the following symmetry relations: Theorem 41 Assume that g(t in (43 and (52 are eual and even functions Then (56 b λ,ν, ( 1 ν c λ 1,ν,, ν > 0 In particular, (57 B NWA,λ,ν, ( 1 ν B JHC,λ 1,ν,, ν > 0 Theorem 42 Assume that g(t in (43 and (52 are eual and even functions Then (58 c (n λ 1,ν, (x ( 1ν λ n b (n λ,ν, (n x This implies a -analogue of the complementary argument theorem [9, p 633, (19] Theorem 43 (59 JHC,λ 1,ν, (x ( 1ν λ n NWA,λ,ν, (n x

10 36 T Ernst Theorem 44 (Compare with [3, 4242] If s l1 n l n, (60 JHC,λ,, (x 1 x s ( m 1,,m s m 1 ++m s where we assume that n j operates on x j s j1 B (n j JHC,λ,m j, (x j, Corollary 45 (Another -analogue of [8, p 300 (49] ( (61 ν JHC,λ,ν, (x B (n 1 JHC,λ,ν, B JHC,λ,,(x Corollary 46 (Another -analogue of [9, p 634, (28, (29] (62 B (n 1 JHC,λ,ν, (x [ 1 ν+1 +1 λ {ν +1} ] JHC,λ,ν+1, (x ( (n 2 B JHC,λ,ν+1, (x Theorem 47 (A -analogue of [6, p 8] B (2n NWA,λ,, (x y(n ν (63 1 λ n ( 1 ν NWA,λ,, (xb(n JHC,λ 1,ν, (y Proof We have that t n (λe (t 1 n E ( t n (xt (λ 1 E 1 ( t 1 n E ( yt (64 t n ( tλe (t n E ((x yt (λe (t 1 n, λe (t 1 which implies that t ν B (2n NWA,λ,ν, (x y (n t l {ν}! {l}! l0 (65 1 t ν NWA,λ,ν, (x ( t m JHC,λ 1,m, (y λ n {ν}! {m}! m0 Formula (63 now follows on euating the coefficients of t ν

11 On certain generalized -Appell polynomial expansions 37 5 The NWA -Apostol Euler polynomials We start with some repetition from [3]: Definition 17 The generating function for the first -Euler polynomials of degree ν and order n, F (n NWA,ν, (x is given by 2 n E (xt (66 (E (t+1 n t ν {ν}! F(n NWA,ν, (x, t <π The following table lists some of the first -Euler numbers F NWA,n, n 0 n 1 n 2 n ( ( n ( 1{3}!( The e polynomials are more general forms of the NWA -Apostol Euler polynomials, which we will study in this section Definition 18 The e polynomials are defined by 2 n g(t (67 (λe (t+1 n E t ν e (n λ,ν, (xt (x {ν}! Definition 19 The generalized NWA -Apostol Euler polynomials (x are defined by F (n NWA,λ,ν, (68 2 n (λe (t+1 n E (xt t ν F (n NWA,λ,ν, (x, t +logλ <π {ν}! Assume that λ 1 The poles in the denominator of (68 are the roots of E (t λ 1 Theorem 51 We have (69 NWA,A, e (n λ,ν, (x e(n 1 λ,ν, (x, This leads to the following recurrence: (70 F NWA,λ,0, 2 λ, λ(f NWA,λ, 1 + F NWA,λ,, 0, >1 The following table lists some of the first F NWA,λ,n, numbers n 0 n 1 n 2 n ( λ 2(λ λ 2λ 2 1+λ (1 + λ 2 (1 + λ 3 (λ +1 4 We observe that the limits for λ 1 are the first -Euler numbers The following two formulas are generalizations of [3, 4202] and [3, 4206]

12 38 T Ernst Theorem 52 (A -analogue of [9, p 635, (31, (32] (71 F (n 1 NWA,λ,ν, (x 1 2 [ (72 x ν 1 λ 2 [ λ ( ] ν F (n NWA,λ,, (x+f(n NWA,λ,ν, (x ( ] ν F NWA,λ,, (x+f NWA,λ,ν, (x Theorem 53 A -analogue of [9, p 636, (43] and a generalization of [3, p 152]: (73 NWA,λ,ν, (x y ( [ ] ν NWA,λ,, (y+{} 2 B(n 1 NWA,λ, 1, (y F NWA,λ,ν, (x Proof We will use the -Taylor formula (15 twice, and then lie in [9], the factor λ disappears (74 NWA,λ,ν, (x y 1 2 by(15, ν ( ν + λ j j0 + λ 2 by( λ 2 j0 NWA,λ,, (y F NWA,λ,j, (x [ F NWA,λ,ν, (x NWA,λ,, (yf NWA,λ,ν,(x j ( ν j j j0 ν j F NWA,λ,j, (x NWA,λ,, (yf NWA,λ,ν,(x ] NWA,λ,, (y NWA,λ,ν j, (y 1F NWA,λ,j, (x by(45 RHS

13 On certain generalized -Appell polynomial expansions 39 Theorem 54 (Another -analogue of [9, p 636, (43] [ ( JHC,λ,ν, (x y JHC,λ,, (y (75 + ( 2 {} 2 B(n 1 JHC,λ, 1, (y ] F NWA,λ,ν, (x ( Proof We will use the -Taylor formula (16 twice, and then lie in [9], the factor λ disappears (76 JHC,λ,ν, (x y 1 2 by(16,72 ν ( ν + λ j j λ 2 j j0 by( λ 2 j j0 ( (n 2 B JHC,λ,, (y ] F NWA,λ,j, (x [ F NWA,λ,ν, (x ( (n 2 B JHC,λ,, (yf NWA,λ,ν,(x j ν j F NWA,λ,j, (x ( (n 2 B JHC,λ,, (yf NWA,λ,ν,(x ( (n 2 B JHC,λ,, (y JHC,λ,ν j, (y 1F NWA,λ,j, (x by(54 RHS Theorem 55 (A -analogue of the addition theorem [6, p 10] (77 NWA,λ 2,ν, (x y (2 n 2 n (2 ν NWA,λ,, (2 xf (n NWA,λ,ν, (2 y Proof We find that 2 n ( n 2 t (2 n λ 2 E ((2 x 2 yt E (2 t 1 (78 ( t n 2 n E (2 xt (λe (t 1 (λe (t+1 n E (2 yt,

14 40 T Ernst which implies that (79 2 n (2 n (2 t ν NWA,λ 2,ν, (x y {ν}! t ν NWA,λ,ν, (2 x {ν}! t m F (n NWA,λ,m, (2 y {m}! m0 Formula (77 now follows on euating the coefficients of t ν 6 The JHC -Apostol-Euler polynomials The f polynomials are more general forms of the JHC -Apostol Euler polynomials, which we will study in this section Definition 20 The f polynomials f (n λ,ν, (x are defined by (80 2 n g(t (λe 1 (t+1 n E (xt t ν f (n λ,ν, (x {ν}! Definition 21 The generalized JHC -Apostol Euler polynomials (x are defined by F (n JHC,λ,ν, (81 2 n (λe 1 (t+1 n E (xt Theorem 61 We have t ν F (n JHC,λ,ν, (x, t +logλ <π {ν}! (82 JHC,A, f (n λ,ν, (x f(n 1 λ,ν, (x, This leads to the following recurrence: (83 F JHC,λ,0, 2 λ, λ(f JHC,λ, 1 + F JHC,λ,, 0, >1 Theorem 62 A generalization of [3, 4224] and another -analogue of [9, p 635, (31]: [ ( ] (84 F (n 1 ν JHC,λ,ν, (x 1 λ ( (n 2 F 2 JHC,λ,ν, (x+f(n JHC,λ,ν, (x Theorem 63 A symmetry relation for -Apostol Euler numbers (85 ( 1 ν F JHC,λ 1,ν, F NWA,λ,ν,

15 On certain generalized -Appell polynomial expansions 41 Proof The following computation with generating functions shows the way: (86 λ ( t ν F JHC,λ 1,ν, {ν}! t ν F NWA,λ,ν, (1 {ν}! 2 λ 1 E 1 ( t+1 Euating the coefficients of t ν and using (70 gives (85 2λE (t λe (t+1 Theorem 64 Assume that g(t in (67 and (80 are eual and even functions Then (87 f (n λ 1,ν, (x ( 1ν λ n e (n λ,ν, (n x This implies a -analogue of the complementary argument theorem [9, p 634, (20] Theorem 65 (88 F (n JHC,λ 1,ν, (x ( 1ν λ n F (n NWA,λ,ν, (n x Theorem 66 (A -analogue of [6, p 8] F (2n NWA,λ,, (x y(n ν (89 1 λ n ( 1 ν Proof We have that 2 n (90 which implies that (91 F (n NWA,λ,, (xf(n JHC,λ 1,ν, (y (λe (t+1 n E (xt (λ 1 E 1 ( t+1 n E ( yt 2 n ( 2λE (t n E ((x yt (λe (t+1 n, λe (t+1 1 λ n t ν F (2n NWA,λ,ν, (x y {ν}! t ν F (n NWA,λ,ν, (x {ν}! l0 2 n (n t l {l}! ( t m F (n JHC,λ 1,m, (y {m}! m0 Formula (89 now follows on euating the coefficients of t ν

16 42 T Ernst 7 More on -Apostol Bernoulli and Euler polynomials It turns out, by simple umbral manipulation that many of the formulas in [3, Section 43] are also valid for -Apostol Bernoulli and Euler polynomials, and these euations which we now present will also be new for the case 1 We first define these polynomials of negative order Definition 22 The generalized NWA -Apostol Bernoulli polynomials of negative order n are given by (92 B ( n NWA,λ,ν, (x {ν}! {ν + n}! n NWA,A, xν+n The generalized NWA -Apostol Euler polynomials of negative order n are given by (93 F ( n NWA,λ,ν, (x n NWA,A, xν Theorem 71 A generalization of [3, 4249]: (94 B ( n p NWA,λ,ν, (x y (B ( n NWA,λ, (x B ( p NWA,λ, (yν, and the same for NWA -Apostol Euler polynomials A special case is the following formula: (95 B ( n NWA,λ,ν, (x y (B ( n NWA,λ, (x y ν, and the same for NWA -Apostol Euler polynomials Theorem 72 A generalization of [3, 4251, 4252]: If n, p Z then (96 B (n+p NWA,λ,ν, (B(n NWA,λ, B (p NWA,λ, ν, and the same for NWA -Apostol Euler numbers Theorem 73 A generalization of [3, 4253]: (97 (x y ν (B ( n NWA,λ, (x NWA,λ, (yν, A generalization of [3, 4254]: (98 (x y ν (F ( n NWA,λ, (x F (n NWA,λ, (yν Proof Put p n in (94 In particular for y 0, we obtain (99 x ν (B ( n NWA,λ, NWA,λ, (xν, (100 x ν (F ( n NWA,λ, F (n NWA,λ, (xν

17 On certain generalized -Appell polynomial expansions 43 This can be restated in the form (101 x ν B ( n (102 x ν s0 s0 NWA,λ,s, {s}! F ( n NWA,λ,s, {s}! D s NWA,λ,ν, (x, D s F (n NWA,λ,ν, (x We conclude that the NWA -Apostol Bernoulli and NWA -Apostol Euler polynomials satisfy linear -difference euations with constant coefficients The following theorem is useful for the computation of NWA -Apostol Bernoulli and NWA -Apostol Euler polynomials of positive order This is because the polynomials of negative order are of simpler nature and can easily be computed When the B ( n NWA,λ,s, to compute the NWA,λ,s, Theorem 74 A generalization of [3, 4259]: (103 s NWA,λ,s, B( n NWA,λ,ν s, δ ν,0 s0 A generalization of [3, 4260]: (104 F (n s NWA,λ,s, F( n NWA,λ,ν s, δ ν,0 s0 Proof Put x y 0in (97 and (98 etc are nown, (103 can be used Theorem 75 (A generalization of [3, 4261] Under the assumption that f(x is analytic with -Taylor expansion (105 f(x D ν f(0 xν {ν}!, we can express powers of NWA,A, and NWA,A, operating on f(x as powers of D as follows These series converge when the absolute value of x is small enough: (106 n NWA,A, f(x D ν+n f(0 B( n NWA,λ,ν, (x {ν}!, (107 n NWA,A, f(x D ν f(0 F( n Proof Use formulas (45 and (54 NWA,λ,ν, (x {ν}!

18 44 T Ernst Putting f(x E (xt we obtain the generating function of the NWA - Apostol Bernoulli and NWA -Apostol Euler polynomials of negative order: (108 (109 (λe (t 1 n t n E (xt (λe (t+1 n 2 n E (xt Theorem 76 (A generalization of [3, 4268] (110 NWA,λ,, (x {}! Proof Replace f(x by f(x y in (45: (111 t ν {ν}! B( n NWA,λ,ν, (x, t ν {ν}! F( n NWA,λ,ν, (x n NWA,A, D f(y D n,xf(x y λf( NWA,λ, (x y 1 f( NWA,λ, (x y D f(b (n 1 NWA,λ, (x y Use the umbral formula [3, 421] to get (112 NWA,λ,, (x {}! NWA,A, D f(y B (n 1 NWA,, (x {}! D +1 f(y Apply the operator n 1 NWA,A, with respect to y to both sides and use (106: (113 NWA,λ,, (x {}! B (n 1 NWA,λ,, (x {}! n NWA,A, D f(y l0 D +l+n f(0 B( n+1 NWA,λ,l, (y {l}! Finally use [3, 421], [3, 434], (97 to rewrite the righthand side Corollary 77 (A generalization of [3, 4272] Let ϕ(x be a polynomial of degree ν A solution f(x of the -difference euation (114 n NWA,A, f(x Dn ϕ(x is given by (115 f(x y NWA,λ,, (x D {}! ϕ(y

19 On certain generalized -Appell polynomial expansions 45 Proof The LHS of (115 can be written as ϕ( NWA,λ, (x y, because if we apply n NWA,,x to both sides we get n NWA,A, f(x y D n,xϕ(x y (116 n NWA,A, ϕ(b(n NWA,λ, (x y Theorem 78 (A generalization of [3, 4275] F (n NWA,λ,, (117 (x n NWA,A, {}! D f(y f(x y Proof Replace f(x by f(x y in (69: 1 ( λf(f (n 2 NWA,λ, (x y 1 + f(f (n NWA,λ, (x y (118 f(f (n 1 NWA,λ, (x y Use the umbral formula [3, 421] to get (119 F (n NWA,λ,, (x {}! NWA,A, D f(y F (n 1 NWA,λ,, (x {}! D f(y Apply the operator n 1 NWA,A, with respect to y to both sides and use (107: F (n NWA,λ,, (x n NWA,A, {}! D f(y (120 F (n 1 NWA,λ,, (x {}! l0 D +l f(0 F( n+1 NWA,λ,l, (y {l}! Finally use [3, 421], [3, 434], (98 to rewrite the righthand side Corollary 79 (A generalization of [3, 4279] Let ϕ(x be a polynomial of degree ν A solution f(x of the -difference euation (121 n NWA,A,f(x ϕ(x is given by F (n NWA,λ,, (122 f(x y (x D {}! ϕ(y Proof The LHS of (122 can be written as ϕ(f (n NWA,λ, (x y, because if we apply n NWA,A,,x to both sides we get (123 n NWA,A, f(x y ϕ(x y n NWA,A, ϕ(f(n NWA,λ, (x y

20 46 T Ernst 8 Multiplicative -Appell polynomials In this section we very briefly discuss multiplicative -Appell polynomials with f n (t eual to h(t n It turns out, by simple umbral manipulation that many of the formulas in [3, Section 43] are also valid for multiplicative -Appell polynomials, and these euations will be presented in another article Throughout, we denote the multiplicative -Appell polynomials by Φ (n M,ν, (x Definition 23 Under the assumption that the function h(t n can be expressed analytically in R[[t]], and for f n (t of the form h(t n, we call the -Appell polynomial Φ in (14 multiplicative (124 E (xth(t n t ν {ν}! Φ(n M,ν, (x, n Z Then we have Theorem 81 If s l1 n l n, n N, (125 Φ (n M,, (x 1 x s m 1 ++m s where we assume that n j operates on x j ( m 1,,m s s j1 Φ (n j M,m j, (x j, Proof Compare with [3, 4242]: In umbral notation we have, lie in the classical case: (126 (x 1 x s n γ ((x 1 n 1 γ (x s n s γ, where γ,,γ are distinct umbrae, each euivalent to γ 9 The -logarithm We wish to investigate the existence of a real inverse for E (x Theorem 91 The function E (x, <x<(1 1 has an inverse Proof It suffices to show that the logarithmic derivative of E (x is > 0 However, this follows from (1 m (127 D (log E (x 1 x(1 m > 0, 0 <<1 m0 Definition 24 The -logarithm log (x is the inverse function of E (x, <x<(1 1, 0 <<1 Theorem 92 The -logarithm log (x has the following properties (x and y have small real values, n N: (1 Its domain is R + ]0, [, and its range is ], (1 1 [ (2 It is strictly increasing

21 On certain generalized -Appell polynomial expansions 47 (3 log (x log (y log (xy (4 log (x log (x log (x log (x n Theorem 93 (A -analogue of [8, p 293 (13] (128 F (n λ,ν, (x E 1 ( x log (λ F (n NWA,+ν, (x(log (λ {}! Proof We euate the generating functions in the following way: (129 t ν F (n λ,ν, (x {ν}! 2 n E 1 ( x log (λ (E (t log (λ + 1 n E (x(t log (λ E 1 ( x log (λ F (n NWA,, (x(t log (λ {}! E 1 ( x log (λ E 1 ( x log (λ m0 F (n NWA,, (x t m {m}! m0 F (n t m (log (λ m { m}!{m}! NWA,+m, (x(log (λ {}! 10 Appendix: multiple -Appell polynomials Of course there are many ways to define multiple -Appell polynomials; in this paper we concentrate on one of the simplest approaches in the spirit of Lee Definition 25 (A -analogue of [7, p 2135] For every power series (130 f(t 1,t 2,l0 a,l t 1 tl 2 {}!{l}!, with f(0, 0 0, the multiple -Appell polynomials Φ,l; (x of degree, l have the following generating function: (131 f(t 1,t 2 E (x(t 1 t 2,l0 t 1 tl 2 {}!{l}! Φ,l;(x Theorem 101 (A -analogue of [7, p 2135] Let {Φ,l; (x},l0 be a multiple -Appell polynomial Then we have euivalently: (1 {Φ,l; (x},l0 is a set of multiple -Appell polynomials

22 48 T Ernst (2 There exists a seuence {Φ,l; },l0 with Φ 0,0; 0such that l ( ( l (132 Φ,l; (x Φ 1,l l l 1 0 l 1,x 1+l 1 1 (3 Let l ( ( l (133 Φ,l;α,β; (x Φ 1,l l l 1 0 l 1,x 1+l 1 α 1+βl 1 1 Then for each n 1 + n 2 1, we have (134 D Φ,l; (x {} Φ 1,l;0,1; (x+{l} Φ,l 1; (x (4 There exists a seuence {Φ,l; },l0 with Φ 0,0; 0such that Φ,l; (x l ( ( (135 l { + l 1 l 1 }! Φ 1,l { + l}! 1 ;D 1+l 1 x +l 1 0 l l 1 Proof (1 (2 This follows since each t 1 and t 2 provides a generating function for a -Appell polynomial (2 (3 By the identity (136 we have (137 { } ( n1 1 ( n2 2 ( 2 n1 1 {n 1 } , 2 1, D Φ,l; (x by( l 1 0 ( n2 2 l ( ( l { 1 + l 1 } 1 ( l ( ( l 1 1 l {} l 1 0 l 1 ( ( l 1 + {l} 1 l l ( ( 1 l {} 1 0 l {l} l l ( ( n1 n2 1 + {n 2 }, l 1 Φ 1,l l 1,x 1+l 1 1 l 1 ( ( l 1 l 1 Φ 1,l l 1,x 1+l 1 1 Φ 1 1,l l 1,x 1+l 1 l 1 Φ 1,l 1 l 1,x 1+l 1 RHS

23 On certain generalized -Appell polynomial expansions 49 The last statement follows after performing the -derivation and changing summation indices 11 Discussion We have discussed six new types of -Appell polynomials, and a multiple form, but the trend is clear; there are many advantages with this general ind of function This theory is very flexible and ready for generalizations In another paper we will prove that the -Appell polynomials (and the Appell polynomials as well form a commutative ring We hope that this article will advertise -Appell polynomials, which are not very well nown; as a coincidence we mention that Paul Appell ( was mathematics teacher of Marie Curie in Paris, they became a very successful team References [1] Apostol, T M, On the Lerch zeta function, Pacific J Math 1 (1951, [2] Dere, R, Simse, Y, Srivastava, H M, A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J Number Theory 133, no 10 (2013, [3] Ernst, T, A comprehensive treatment of -calculus, Birhäuser, Basel, 2012 [4] Ernst, T, -Pascal and -Wronsian matrices with implications to -Appell polynomials, J Discrete Math 2013 [5] Jordan, Ch, Calculus of finite differences, Third Edition, Chelsea Publishing Co, New Yor, 1950 [6] Kim M, Hu S, A note on the Apostol Bernoulli and Apostol Euler polynomials, Publ Math Debrecen 5587 (2013, 1 16 [7] Lee, D W, On multiple Appell polynomials, Proc Amer Math Soc 139, no 6 (2011, [8] Luo, Q-M, Srivastava, H M, Some generalizations of the Apostol Bernoulli and Apostol Euler polynomials, J Math Anal Appl 308, no 1 (2005, [9] Luo, Q-M, Srivastava, H M, Some relationships between the Apostol Bernoulli and Apostol Euler polynomials, Comput Math Appl 51, no 3 4 (2006, [10] Luo, Q-M, Apostol Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J Math 10, no 4 (2006, [11] Milne-Thomson, L M, The Calculus of Finite Differences, Macmillan and Co, Ltd, London, 1951 [12] Nørlund, N E, Differenzenrechnung, Springer-Verlag, Berlin, 1924 [13] Pintér, Á, Srivastava, H M, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aeuationes Math 85, no 3 (2013, [14] Sandor, J, Crstici, B, Handboo of number theory II, Kluwer Academic Publishers, Dordrecht, 2004 [15] Srivastava, H M, Özarslan, M A, Kaanoglu, C, Some generalized Lagrange-based Apostol Bernoulli, Apostol Euler and Apostol Genocchi polynomials, Russ J Math Phys 20, no 1 (2013, [16] Wang, W, Wang, W, Some results on power sums and Apostol-type polynomials, Integral Transforms Spec Funct 21, no 3 4 (2010, [17] Ward, M, A calculus of seuences, Amer J Math 58 (1936,

24 50 T Ernst Thomas Ernst Department of Mathematics Uppsala University PO Box 480, SE Uppsala Sweden thomas@mathuuse Received February 1, 2014