EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES

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1 LE MATEMATICHE Vol. LXXIII 208 Fasc. I, pp doi: 0.448/ EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES THOMAS ERNST We preset idetities of various kids for geeralized Apostol- Beroulli ad Apostol-Euler polyomials ad power sums, which resemble aalogues of formulas from the 2009 paper by Liu ad Wag. These formulas are divided ito two types: formulas with oly Apostol- Beroulli, ad oly Apostol-Euler polyomials, or so-called mixed formulas, which cotai polyomials of both kids. This ca be see as a logical coseuece of the fact that the Appell polyomials form a commutative rig. The fuctioal euatios for Ward umbers operatig o the expoetial fuctio, as well as symmetry argumets, are essetial for may of the proofs. We coclude by fidig multiplicatio formulas for two Appell polyomials of geeral form. This brigs us to the H polyomials, which were discussed i a previous paper.. Itroductio I the secod article o aalogues of two Appell polyomials [4], the Apostol- Beroulli ad Apostol-Euler polyomials, focus was o multiplicatio formulas ad o formulas icludig multiple λ power sums. I this article we will fid a correspodig multiplicatio formula for a more geeral Appell polyomial, which is a geeralizatio of both Apostol-Euler ad Apostol-H polyomials. Etrato i redazioe: 4 settembre 207

2 4 THOMAS ERNST There are may ew formulas o this subject, both Apostol-Appell ad similar Appell, which have recetly bee published; i all cases the limit λ is straightforward. Sometimes we write -aalogue of etc., ot botherig about the above dichotomy. This paper is orgaized as follows: I sectio we give a geeral itroductio och the defiitios. I sectio 2 we preset formulas with oly Apostol- Beroulli, ad oly Apostol-Euler polyomials. I sectio 3 we preset mixed formulas for these polyomials. I sectio 4, two geeral polyomials are defied, which geeralize the Apostol-Beroulli ad Apostol-Euler polyomials. The multiplicatio formulas for these polyomials are proved, which specialize to the Apostol-H polyomials. We ow start with the defiitios. Some of the otatio is well-kow ad ca be foud i the book []. The variables i, j,k,l,m,, vill deote positive itegers, ad λ,µ will deote complex umbers whe othig else is stated. Defiitio.. The Gauss biomial coefficiet are defied by {}!,k 0,,...,. k {k}!{ k}! Let a ad b be ay elemets with commutative multiplicatio. The the NWA additio is give by a b k0 a k b k, 0,,2,... 2 k If 0 < < ad z <, the expoetial fuctio is defied by E z k0 {k}! zk. 3 The followig theorem shows how Ward umbers usually appear i applicatios. Theorem.. Assume that,k N. The k k m +...+m k m,...,m, 4 where each partitio of k is multiplied with its umber of permutatios. Theorem.2. Fuctioal euatios for Ward umbers operatig o the expoetial fuctio. First assume that the letters m ad are idepedet,

3 ON Q-APPELL POLYNOMIALS 5 i.e. come from two differet fuctios, whe operatig with the fuctioal. Furthermore, mt <. The we have E m t E m t. 5 Furthermore, E jm E j m E m j. 6 Compare with the semirig of Ward umbers [, p. 67]. Proof. Formula 5 is proved as follows: E m t E t, 7 where the umber of s to the left is m. But this meas exactly E t m, ad the result follows. Defiitio.2. The geeralized NWA Apostol-Beroulli polyomials x are defied by B NWA,λ,, t λe t E xt t B NWA,λ,, x {}!, t + logλ < 2π. 8 Defiitio.3. The geeralized NWA Apostol-Euler polyomials x are defied by F NWA,λ,, 2 λe t + E xt t F NWA,λ,, x {}! Defiitio.4. The geeralized NWA H polyomials are defied by 2t λe t + E xt t H NWA,λ,, x {}! Defiitio.5. The geeralized JHC H polyomials are defied by 2t λe t + E xt t H, t + logλ < π. 9, t + logλ < π. 0 JHC,λ,, x, t + logλ < π. {}!

4 6 THOMAS ERNST Defiitio.6. The geeratig fuctio for H NWA,, x is give by 2t E t + E xt t H NWA,, x {}!, t < 2π. 2 Defiitio.7. The geeratig fuctio for H JHC,, x is give by E 2t t + E xt t H JHC,,, x {}!, t < 2π. 3 The polyomials i 2 ad 3 are aalogues of the geeralized H polyomials. Defiitio.8. The polyomials b λ,, x are defied by t gt λe t E xt Defiitio.9. The e polyomials are defied by 2 gt λe t + E xt t b λ,, x {}! t e λ,, x {}! The f polyomials are more geeral forms of the JHC H polyomials. Defiitio.0. The f polyomials f λ,, x are defied by 2 gt λe t + E xt t f λ,, x {}!. 6 Defiitio.. A aalogue of [7, 20 p. 38], the multiple power sum is defied by s l l NWA,λ,m, j λ k m k, 7 j l where k j + 2 j j, j i 0. Defiitio.2. A aalogue of [7, 46 p. 386], the multiple alteratig power sum is defied by σ l l NWA,λ,m, l j λ k m k, 8 j l where k j + 2 j j, j i 0.

5 ON Q-APPELL POLYNOMIALS 7 Theorem.3. A symmetry relatio for the geeralized H umbers. H JHC,λ,, H NWA,λ,,. 9 Proof. A simple computatio with geeratig fuctios shows the way: t H JHC,λ,, {}! λ t H NWA,λ,,. {}! 2t λ E t + 2tλE t λe t + 20 Euatig the coefficiets of t gives 9. Theorem.4. Assume that gt i 5 ad 6 are eual ad eve fuctios. The f λ,, x λ e λ,, x. 2 This implies a complemetary argumet theorem for the geeralized H polyomials. Theorem.5. H JHC,λ,, x λ H NWA,λ,, x, eve. 22 H JHC,λ,, x + λ H NWA,λ,, x, odd. 23 Defiitio.3. The followig fuctios amed the power sum, ad the alterate power sum with respect to λ, were itroduced i [4]. s NWA,λ,m, k0 λ k k m ad σ NWA,λ,m, Their respective geeratig fuctios are k0 k λ k k m. 24 m0 t m s NWA,λ,m, {m}! λ E t λe t 25 ad m0 t m σ NWA,λ,m, {m}! + λ E t λe t +

6 8 THOMAS ERNST 2. The first expasio formulas Theorem 2.. A triple sum of NWA Apostol-Euler polyomials is eual to aother triple sum of NWA Apostol-Euler polyomials. i j 2 F k NWA,λ i j, x F k, NWA,λ j i, 2 y σ, NWA,λ j, 3,i j 3 F k NWA,λ i,, i j F k NWA,λ j,, j x jm i. i y i λ jm m m0 27 Proof. Defie the followig fuctio, ote that f t is symmetric whe i, j have the same parity. f t E i j x yt i+ λ i j E i j t + λ i E i t + k λ j E j t + k 2 2k E i j x yt 2 k k 2 i+ λ i j E i j t + λ i E i t + λ j E j t + λ j. E j t + 28 By usig the formula for a geometric seuece, we ca expad f t i two ways: f t by26,9 2 2k F k NWA,λ i j x i t σ,, {}! NWA,λ j,m,i j t m m0 {m}! i y j t l 2 2k 2 k λ i E i t + k F k NWA,λ j,l, l0 2 k λ j E j t + k 2 i 2k m0 i m0 {l}! m λ jm m λ jm E l0 j t 0 {}! F k NWA,λ j,, i y. j x j y jm i l t l {l}! F k jm j NWA,λ i,l, x i The theorem follows by euatig the coefficiets of i t {}!. i t 29

7 ON Q-APPELL POLYNOMIALS 9 Theorem 2.2. Almost a aalogue of [5, p. 335]. Assume that i ad j are either both odd, or both eve. The we have j i F k NWA,λ i,, i j F k NWA,λ j,, i λ jm m F k NWA,λ i,, m0 j y j λ im m F k im i NWA,λ j,, x m0 j i y j x jm i 30 Proof. This follows from the previous proof, ad the usig the symmetry for i ad j. Theorem 2.3. A triple sum of NWA Apostol-Beroulli polyomials is eual to a double sum of NWA Apostol-Beroulli polyomials. i j 2 j 3 B k NWA,λ i,, i j B k NWA,λ j,, j x B k NWA,λ j, 2, i y s NWA,λ j, 3,i i y i λ jm B k jm j NWA,λ i,, x m0 i 3 Proof. Defie the followig symmetric fuctio φ t E i j x ytλ i j E i j t λ i E i t k λ j E j t k tk E i j x yt k k i t j t λ i j E i j t t 2k λ i E i t λ j E j t λ j E j t i k j k. 32 By usig the formula for a geometric seuece, we ca expad φ t i two

8 0 THOMAS ERNST ways: φ t by25 B k NWA,λ j,l, l0 j t k λ j E j t k t 2k i k j k B k NWA,λ i j x i t,, i y j t l {l}! j t 0 {}! Bk NWA,λ j,, i y. s {}! NWA,λ j,m,i j t m m0 {m}! t 2k i k j k i t k λ i E i t k i λ jm jm E j x j y m0 i i λ jm i l t l m0 l0 {l}! Bk NWA,λ i,l, The theorem follows by euatig the coefficiets of j x jm i t {}!. Theorem 2.4. A aalogue of [2, p. 2994], [, p. 55]. i j 2 j 3 B k NWA,λ i,, j i 2 i 3 B k NWA,λ j,, Proof. Use the symmetry i φ t. t 2k i t i k j k 33 j x B k NWA,λ j, 2, i y s NWA,λ j, 3,i i x B k NWA,λ i, 2, j y s NWA,λ i, 3, j 34 Theorem 2.5. A aalogue of [2, p. 2996]. We have i j λ l+m i j B k NWA,λ,, l0 m0 B k NWA,λ,, j i l0 m0 B k NWA,λ,, i y im j λ l+m j i B k NWA,λ,, j y jm i j x jl i il i x j 35

9 ON Q-APPELL POLYNOMIALS Proof. We ca expad the followig symmetric fuctio φ t by usig the formula for a geometric seuece: φ t E i j x ytλ i E i j t λ j E i j t λe i t k λe j t k t 2k 2 E i j x yt i k j k k k i t j t λ i E i j t λ j E i j t λe i t λe j t λe j t λe i t i j k k i k j k λ l+m i t j t l0 m0 λe i t λe j t jl im E j x i t E i y j i j t i k j k j λ m m0 2 0 i λ l l0 j 2t 2 0 { 2 }! Bk NWA,λ, 2, i t { }! Bk NWA,λ,, i y im j. j x jl i 36 The theorem follows by usig the symmetry i φ t ad chagig k to k. Theorem 2.6. A aalogue of [2, p. 2997]. We have i l0 j λ l+m B k NWA,λ,, m0 i j B k j l0 i λ l+m B k NWA,λ,, m0 NWA,λ,, j x jl i m j i B k i y NWA,λ,, il i x m. j j y 37 Proof. Similar to above.

10 2 THOMAS ERNST Theorem 2.7. A aalogue of [, p. 552]. We have i k j k i m j m m m0 B k NWA,λ j, m, j k i k B k NWA,λ i, m, i y i m0 l0 m λ jl B k NWA,λ i,m, j m i m j y j λ il B k NWA,λ j,m, l0 j x jl i i x il j. 38 Proof. We ca expad the followig symmetric fuctio ψ t by usig the formula for a geometric seuece: ψ t E i j x ytλ i j E i j t λ i E i t k λ j E j t k t2k E i j x yt i k j k k k i t j t λ i j E i j t λ i E i t λ j E j t λ j E j t k k i t j t i i k j k λ i E i t λ j E λ l j j t l0 jl E j x i t E i y j t i i i k j k l0 λ jl 0 j 2t { 2 }! Bk NWA,λ j i, 2 y, λ jl i m j m B k NWA,λ i,m, i t jl { }! Bk NWA,λ i, j, x i k j k j x jl i 0 m0 i m i l0 B k NWA,λ j i y t, m, {}!. 39 The theorem follows by usig the symmetry i ψ t. 3. Mixed formulas This is a cotiuatio of the very similar computatios i [4], to which we will refer.

11 ON Q-APPELL POLYNOMIALS 3 Corollary 3.. A aalogue of [0, 3 p. 34]. If i is eve the λ im im F NWA,λ 2,, i x m0 2 2 i k {} 2 k0 k 2 k k0 2 i {} 2 i i 2 k B NWA,λ i,k, 2 x σ NWA,λ 2, k,i 2 k i k F NWA,λ 2,k, i x s NWA,λ i, k,2 i m0 m λ 2m B NWA,λ i,, 2m 2 x. i Proof. Put j 2 i formula 56 [4], ad multiply by 2 {} Corollary 3.2. A aalogue of [0, 32 p. 34]. m0 m+ λ m 2m B NWA,λ,, x {} λ m 2m F NWA,λ,, x m Proof. Put i 2 i formula 40, replace x ad λ 2 by x 2 ad λ, ad multiply by {} 2 2. Corollary 3.3. A aalogue of [0, 33 p. 34]. m0 m λ jm B NWA,λ 2,, m0 j x jm 2 {} 2 k0 j k 2 k F NWA,λ j,k, 2 x s NWA,λ 2, k, j {} 2 j j λ 2m 2m F NWA,λ j,, 2 x. j k 42 Proof. Put i 2 i formula 56 [4], ad multiply by 2 2. The followig formula is a geeralizatio of [4, 57].

12 4 THOMAS ERNST Theorem 3.. A aalogue of [5, 3.9 p. 3356]. B k NWA,λ i,, i j 2 B k NWA,λ i,, i j F k NWA,λ j,, j x jm i Proof. Defie the followig fuctio. j x F k NWA,λ j i, 2 y σ, NWA,λ j, 3,i j 3 i y i λ jm m m0 g t E i j x yt i+ λ i j E i j t + λ i E i t k λ j E j t + k 2 k i t k E i j x yt k k i t 2 i+ λ i j E i j t + λ i E i t λ j E j t + λ j. E j t By usig the formula for a geometric seuece, we ca expad g t i two ways: g t by26 2 k i t k F k NWA,λ j,l, l0 2 k λ j E j t + k 2 k i t k i m0 B k NWA,λ i j x i t,, {}! m0 i y j t l i m0 m λ jm 2 k {l}! i t k m λ jm E l0 j t 0 {}! F k NWA,λ j,, i y. i t λ i E i t σ NWA,λ j,m,i j t m {m}! k j x j y jm i l t l jm {l}! Bk j NWA,λ i,l, x i The theorem follows by euatig the coefficiets of i t {}!. i t 45

13 ON Q-APPELL POLYNOMIALS 5 Theorem 3.2. A aalogue of [5, p. 3353]. Uder the assumptio that i is eve, we have i j 2 B k NWA,λ i j, x F k, NWA,λ j i, 2 y s, NWA,λ j, 3,i j 3 j y {} i k 2i k F k NWA,λ j, 2, i x s NWA,λ i, 3, j. i j 2 j 3 B k NWA,λ i,, 46 Proof. We ca write g t as follows: g t by25,44 F k NWA,λ j,l, l0 2 λ j E j t + 2 k 2 k i t k B k NWA,λ i j x i t s,, {}! NWA,λ j,m,i j t m m0 {m}! k 2 k {l}! i t k E i t i j x yt λ i E i t by25 λ i E i t i y j t l k λ i j E i j t F k NWA,λ j,, i x j t {}! i t k s NWA,λ i,m, j j t m B k m0 {m}! NWA,λ i,l, l0 j y i t l. {l}! 47 The theorem follows by euatig the coefficiets of t {}!. Theorem 3.3. A aalogue of [5, p. 3353]. Uder the assumptio that i is

14 6 THOMAS ERNST eve, i j F k NWA,λ j i y i,, m0 B k NWA,λ i,, {} i k 2i k F k NWA,λ j,k, j x jm k0 i k i x im j Proof. We ca expad g t as follows: g t by44 2 k i t k E i j x yt i t λ i E i t λ jm m i k j k B k NWA,λ i, k,. j y j λ im m0 k k 2 λ i j E i j t λ j E j t + λ i. E i t 48 2 k i t k 2 k λ j E j t + k 2 k i t k i t λ i E i t k j λ im im E i x m0 j j t j λ im i l t l m0 l0 {l}! Bk NWA,λ i j y,l, j t 0 {}! F k NWA,λ j,, i im x. j The theorem follows by euatig the coefficiets of Theorem 3.4. i j 2 j 3 F k NWA,λ i,, i j B k NWA,λ j,, t {}!. E i j yt 49 j x B k NWA,λ j, 2, i y s NWA,λ j, 3,i i y i m0 λ jm F k NWA,λ i,, j x jm i 50

15 Proof. Defie the followig fuctio ON Q-APPELL POLYNOMIALS 7 Ψ t E i j x ytλ i j E i j t λ i E i t + k λ j E j t k tk E i j x yt 2 k k j t λ i j E i j t 2 k λ i E i t + λ j E j t λ j E j t j k. 5 By usig the formula for a geometric seuece, we ca expad Ψ t i two ways: Ψ t by25 F k NWA,λ i j x i t s,, NWA,λ j,m,i j t m {m}! B k NWA,λ j,l, l0 j t k λ j E j t k i m0 2 k λ jm j k i y j t l {l}! {}! m0 2 k j k 2 k λ i E i t k i λ jm jm E j x j y m0 i i l t l l0 {l}! F k NWA,λ i,l, j t 0 {}! Bk NWA,λ j,, i y. j x jm i The theorem follows by euatig the coefficiets of t {}!. i t 2 k j k 52 The followig example illustrates that similar formulas with H polyomials ca easily be costructed. Theorem 3.5. i j 2 j 3 H k NWA,λ i,, i j B k NWA,λ j,, Proof. Use Ψ t agai. Theorem 3.6. A aalogue of [5, 3. p. 3356]. j x B k NWA,λ j, 2, i y s NWA,λ j, 3,i i y i λ jm H k jm j NWA,λ i,, x m0 i 53

16 8 THOMAS ERNST i j 2 j 3 F k NWA,λ i,, i j B k NWA,λ j,, Proof. Defie the followig fuctio j x B k NWA,λ j, 2, i y NWA,λ j, 3,i i y i λ jm F k jm j NWA,λ i,, x m0 i 54 f t E i j x ytλ i j E i j t λ i E i t + k λ j E j t k tk E i j x yt 2 k k j t λ i j E i j t λ i E i t + λ j E j t λ j E j t 2 k j k. 55 By usig the formula for a geometric seuece, we ca expad f t i two ways: f t by25 B k NWA,λ i j x i t,, B k NWA,λ j,l, l0 j t k λ j E j t k i m0 λ jm 2 k j k i y j t l {l}! s {}! NWA,λ j,m,i j t m m0 {m}! 2 k j k 2 k λ i E i t + k i λ jm jm E j x j y m0 i i l t l l0 {l}! F k NWA,λ i,l, j t 0 {}! Bk NWA,λ j,, i y. j x jm i The theorem follows by euatig the coefficiets of t {}!. i t 2 k j k Multiplicatio formulas We will ow defie two uite geeral Appell polyomials, which have some similarities with the Appell polyomials i [9]. The ames are chose to resemble the Euler ad Beroulli polyomials.

17 ON Q-APPELL POLYNOMIALS 9 Defiitio 4.. A aalogue of Lu, Luo [6, p. 4]. The geeratig fuctio for the geeralized NWA Apostol E polyomials of degree ad order, E NWA,λ,µ,θ;, x, is give by 2 µ t θ t E xt λe t + {}! E NWA,λ,µ,θ;, x,θ N. 57 Several Appell polyomials i this article are special cases of these polyomials, e.g. the Euler polyomial is the case θ 0, µ. Theorem 4.. A aalogue of [6, 2.3 p. 5], first multiplicatio formula for Apostol-E polyomials E NWA,λ,µ,θ;, m x m m θ λ k E k j NWA,λ m,µ,θ;, x, m j 58 where k j + 2 j m j m, m odd. Proof. E NWA,λ,µ,θ;, m x t {}! m 2 µ t θ λ m E m t + by6 2 µ t θ m θ λ m E m t + m m θ j i0 2 µ t θ λe t + E m xt λ i E i t E m xt j j λ k k E x m t m j λ k E NWA,λ m,µ,θ;, m θ k x t m {}!. 59 The theorem follows by euatig the coefficiets of t {}!. The followig formula oly applies for special values of the itegers. Theorem 4.2. A aalogue of [6, 2.4 p. 5], secod multiplicatio formula for Apostol-E polyomials. E NWA,λ,µ,θ;, m x 2 µ m + θ { + } θ, m λ k j j B NWA,λ m,+ θ, k x, m 60

18 20 THOMAS ERNST where k j + 2 j m j m, m eve, θ. Proof. E NWA,λ,µ,θ;, m x t {}! m 2 µ t θ λe t + E m xt 2 µ t θ λ m E m t λ i E i t E m xt i0 2 µ tm λ m E m t j λ k k t θ E x m t m m j t θ 2 µ m m j λ k B k NWA,λ m,; x t m {}! j 6 The theorem follows by euatig the coefficiets of t {}!. Corollary 4.2. A aalogue of [8, 2. p. 49], [6, p. 7], first multiplicatio formula for geeralized H polyomials. H NWA,λ,, m x m m λ k H k j NWA,λ m,, x, m 62 j where k j + 2 j m j m, m odd. Corollary 4.3. A aalogue of [8, 2.2 p. 49], [6, p. 7], secod multiplicatio formula for geeralized H polyomials. H NWA,λ,, m x 2 m m λ k j B k NWA,λ m,, x, m j 63 where k j + 2 j m j m, m eve. Theorem 4.3. A aalogue of [8, p. 5], a explicit formula for the multiple alteratig power sum: σ l l NWA,λ,, 2 l +l m0 + l m j0 H j NWA,λ,m, l j λ j+l j { + } l, j + l H l j NWA,λ,+l m,m,, 64

19 ON Q-APPELL POLYNOMIALS 2 Proof. We use the geeratig fuctio techiue. Put k j + 2 j j. It is assumed that j i 0, i. All zeros are eglected. σ l NWA,λ,, t by7 l {}! l j λ k k t {}! j l λe t λ 2 E 2 t + + λ E t l λ E t λe t + + λe l t λe t + l l λ E t j λe t l j j0 j λe t + λe t + by6 2t l l l j λ j+l j0 j H j t m NWA,λ,m, j + l m0 {m}! [ l j t i H NWA,λ,i, 2 l l l j λ j+l j { + } l, i0 +l m0 + l m {i}! j0 j + l H H j NWA,λ,m, l j NWA,λ,+l m,m, ] t {}!. 65 The theorem follows by euatig the coefficiets of t {}!. Theorem 4.4. For m odd, we have the followig recurrece relatio for Apostol-E -umbers. E l NWA,λ,µ,θ;, m l l j0 j m θ l E m j NWA,λ m,µ,θ; j, σ l 66 where k j + 2 j m j m i σ l NWA,λ, j, m. Proof. E l by58 NWA,λ,µ,θ;, m m θ l l λ k l j0 m m θ l m j m j m θ l E l j0 l λ k E l k NWA,λ m,µ,θ;, m l j k j E l NWA,λ m,µ,θ; j, NWA,λ m,µ,θ; j, l m l λ k NWA,λ, j, m, j by7 k LHS. 67

20 22 THOMAS ERNST Defiitio 4.4. The geeratig fuctio for the geeralized NWA Apostol C polyomials of degree ad order, C NWA,λ,θ;, x, is give by t θ t E xt λe t {}! C NWA,λ,θ;, x,θ N. 68 Theorem 4.5. Multiplicatio formula for Apostol-C polyomials C NWA,λ,θ;, m x m m θ where k j + 2 j m j m. Proof. λ k C k j NWA,λ m,θ;, x, m 69 j C NWA,λ,θ;, m x t θ t {}! m λ m E m t by6 t θ m θ λ m E m t m m θ j i0 t θ λe t E m xt λ i E i t E m xt j jλ k k E x m t m j λ k C NWA,λ m,θ;, m θ k x t m {}!. 70 The theorem follows by euatig the coefficiets of t {}!. 5. Discussio This was the first multiplicatio formula for a Appell polyomial of geeral form; the Ward umbers replace the itegers i the fuctio argumet. Certaily there are other geeral Appell polyomials with similar expasio ad multiplicatio formulas. May of the proofs use the formula for a geometric seuece i form ad the geeratig fuctio for the Appell polyomials ad the power sums. The itegers i ad j are crucial for the formulas; by the geeratig fuctio, if λ i, appears as idex i a polyomial, certaily the factor i will also appear. If the orders of two polyomials i a formula are k ad k, the last oe with

21 ON Q-APPELL POLYNOMIALS 23 idex λ j, ad argumet i y, surely a fuctio σ NWA,λ j,m,i or s NWA,λ j,m,i, together with j m will appear. If a polyomial has λ i, as idex, it will have j i the fuctio argumet, ad vice versa. These cosideratios also hold for the case. Eve if the reader is ot iterested i calculus, this paper is a good summary of the recet treds o Apostol type Appell polyomials; just put. REFERENCES [] Erst.T., A comprehesive treatmet of calculus, Birkhäuser 202. [2] Erst, T. O certai geeralized Appell polyomial expasios A. Uiv. Marie Curie, Sect. A 68, No. 2, [3] Erst, T. A solid foudatio for Appell Polyomials. ADSA 0, [4] Erst, T. O multiplicatio formulas for Apostol-Beroulli ad Apostol- Euler polyomials ad the multiple power sums, A. Uiv. Marie Curie, Sect. A 70, No., [5] Liu, Hogmei; Wag, Weipig Some idetities o the Beroulli, Euler ad Geocchi polyomials via power sums ad alterate power sums. Discrete Math , o. 0, [6] Lu, D., Luo, Q.: Some uified formulas ad represetatios for the Apostol-type polyomials. Advaces i Differece Euatios 205:37 [7] Luo, Q.-M. The multiplicatio formulas for the Apostol-Beroulli ad Apostol- Euler polyomials of higher order. Itegral Trasforms Spec. Fuct , o. 5-6, [8] Luo, Q.-M Multiplicatio formulas for Apostol-type polyomials ad multiple alteratig sums. A traslatio of Mat. Zametki 9 202, o., Math. Notes 9 202, o. -2, [9] T.R. Prabhakar ad Reva, A Appell cross-seuece suggested by the Beroulli ad Euler polyomials of geeral order. Idia J. Pure Appl. Math. 0 o , [0] Wag, Weipig; Wag, Wewe Some results o power sums ad Apostol-type polyomials. Itegral Trasforms Spec. Fuct , o. 3-4, [] Yag, Sheg-liag A idetity of symmetry for the Beroulli polyomials. Discrete Math , o. 4, [2] Zhag, Zhizheg; Yag, Haig Several idetities for the geeralized Apostol- Beroulli polyomials. Comput. Math. Appl , o. 2,

22 24 THOMAS ERNST THOMAS ERNST Departmet of Mathematics Uppsala Uiversity P.O. Box 480, SE Uppsala, Swede