SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS RELATED TO GENERALIZED POWER AND ALTERNATING SUMS

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1 Bullet of Mthemtcl Alyss d Applctos ISSN: , URL: Volume 4 Issue 4 01, Pges SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS RELATED TO GENERALIZED POWER AND ALTERNATING SUMS COMMUNICATED BY R.K. RAINA B.-J. FUGÈRE, S. GABOURY, R. TREMBLAY Astrct. The purpose of ths pper s to ot severl seres dettes volvg some clsses of geerlzed Apostol-Beroull d Apostol-Euler polyomls troduced ltely y Srvstv et l. 16, 17] s well s the geerlzed sum of teger powers, the geerlzed ltertg sum d the logues of the expsos of the hyperolc tget d the hyperolc cotget. The method used s tht of geertg fuctos. It c e foud tht my dettes recetly oted re specl cses of our results. 1. Itroducto, Deftos d Nottos The geerlzed Beroull polyomls B α x of order α C, the geerlzed Euler polyomls E α x of order α C d the geerlzed Geocch polyomls G α x of order α C, ech of degree s well s α, re defed respectvely y the followg geertg fuctos see,4, vol.3, p.53 et seq.], 8, Secto.8] d 10]: α t e t e xt B α 1 xt t < π; 1 α : 1, 1.1! 0 α e t e xt + 1 xt t < π; 1 α : 1 1.! d α t e t e xt G α xt! t < π; 1 α : The lterture cots lrge umer of terestg propertes d reltoshps volvg these polyomls 1,, 3, 4, 5, 15]. Q.-M. Luo d Srvstv 1, 14] troduced the geerlzed Apostol-Beroull polyomls B α x of order α, 1991 Mthemtcs Suject Clssfcto. Prmry 11B68; Secodry 11S80. Key words d phrses. Beroull umers d polyomls, Euler umers d polyomls, Geocch umers d polyomls, Apostol-Beroull polyomls, Apostol-Euler polyomls, Apostol-Geocch polyomls, Geerlzed power sums, Geerlzed ltertg sums. Sumtted Jue 19, 01. Pulshed Novemer 6,

2 Q.-M. Luo 9] vestgted the geerlzed Apostol-Euler polyomls x of order α d the geerlzed Apostol-Geocch polyomls G α x of order α see lso,10, 11, 13]. The geerlzed Apostol-Beroull polyomls B α x; λ of order α C, the geerlzed Apostol-Euler polyomls x; λ of order α C, the geerlzed Apostol-Geocch polyomls G α x; λ of order α C re defed respectvely y the followg geertg fuctos α t λe t e xt B α x; λt t + l λ < π; 1 α : ! d 0 α λe t e xt + 1 α t λe t e xt + 1 It s esy to see tht 0 0 G α x; λt! x; λt! 77 t + l λ < π; 1 α : t + l λ < π; 1 α : B α x B α x; 1, E α x x; 1 d G α x G α x; 1. Recetly, Srvstv et l. 16, 17] hve vestgted some ew clsses of Apostol-Beroull, Apostol-Euler d Apostol-Geocch polyomls wth prmeters, d c defed y the followg geertg fuctos. Defto 1.1. Let,, c R +, d N 0. The geerlzed Apostol- Beroull polyomls B α x; λ;,, c of order α, the geerlzed Apostol-Euler polyomls x; λ;,, c of order α d the geerlzed Apostol-Geocch polyomls G α x; λ;,, c of order α re defed respectvely y the followg geertg fuctos α t λ t t c xt B α x; λ;,, c t t!, l + l λ < π; 1 α : 1, 1.7 α λ t + t c xt x; λ;,, c t t!, l + l λ < π; 1 α : d α t λ t + t c xt G α x; λ;,, c t t!, l + l λ < π; 1 α : If we te 1, c e 1.7, 1.8 d 1.9 respectvely, we hve 1.4, 1.5 d 1.6. Ovously, whe we set λ 1, α 1, 1, c e 1.7, 1.8 d 1.9, we hve clsscl Beroull polyomls B x, clsscl Euler polyomls E x d clsscl Geocch polyomls G x.

3 78 For ech N 0, S defed y S j 1.10 s clled the sum of teger powers. The expoetl geertg fucto for S s gve y 19] S t! e+1t 1 e t We ow defe the geerlzed sum of teger powers s follows. Defto 1.. For rtrry rel or complex λ, the geerlzed sum of tegers powers S ; λ s defed y the geertg relto S ; λ t! λe+1t 1 λe t It s ovous tht 0 For N 0 d N, T defed y s clled the ltertg sum. gve y 0 S ; 1 S T The expoetl geertg fucto for T s T t! 1 1 e t 1 + e t The geerlzed ltertg sum of order α s defed 7] s follows. Defto 1.3. For y rtrry rel or complex prmeter λ, the geerlzed ltertg sum of order α, T α ; λ s defed y the followg geertg fucto: T α ; λ t 1 λ 1! e t α 1 + λe t It s esy to oserve tht T 1 ; 1 T I ths pper, we preset severl seres dettes volvg the geerlzed Apostol- Beroull d the geerlzed Apostol-Euler polyomls defed respectvely y 1.7 d 1.8. I Secto, we ot severl symmetry dettes for the geerlzed Apostol-Beroull polyomls relto etwee the these polyomls d the geerlzed sum of teger powers 1.1. I Secto 3, we prove severl dettes volvg the geerlzed Apostol-Euler, the geerlzed ltertg sum d the logues of the expsos of the hyperolc tget d the hyperolc cotget. Some dettes re lso oted y usg the reltoshps etwee the geerlzed Apostol-Beroull, Apostol-Euler d Apostol-Geocch polyomls.

4 79. Symmetry dettes for the geerlzed Apostol-Beroull polyomls I ths secto, we estlsh some symmetry dettes volvg the geerlzed Apostol-Beroull polyomls B α x; λ;,, c defed the frst secto d the geerlzed sum of teger powers defed y 1.1. Ths s doe y usg the method of geertg fuctos. These results provdes geerlzto of ow dettes 18, 1,, 3] Theorem.1. For ll tegers µ > 0, ν > 0, α 1, 0, for,, c R + d for λ C, we hve the followg detty: B α νx + µ 1ν l ; λ;,, c µ ν +1 S µ 1, λ B α 1 µy; λ;,, c l B α µx + ] ν 1µ l ; λ;,, c ν µ +1 S ν 1, λ B α 1 νy; λ;,, c l ]..1 Proof. Cosderg gt tα 1 c µνxt λ µνt µνt λ µt µt α λ νt νt α. We hve to expd the lst fucto to seres two wy to prove the theorem. We hve gt tα 1 c µνxt λ µνt µνt λ µt µt α λ νt νt α α 1 µt c νxµt λ µνt µνt α 1 νt µ α ν α 1 λ µt µt λ νt νt λ νt νt c νt l α l c µt c νxµt λ ν µt 1 α 1 µ α ν α 1 λ µt µt ν λ νt ν t 1 λ νt νt ν 1 µ α ν α 1 S µ 1, λ 0 1 µ α ν α B α νx ν l B α µ 1ν l µt ; λ;,, c! ] t νx +! 0 B α 1 µy; λ;,, c νt! µ 1ν l ; λ;,, c S µ 1, λ B α 1 µy; λ;,, c l µ ν +1 ] ] t!..

5 80 By expdg dfferet wy, we hve gt 1 B α ν α µ α µx S ν 1, λ B α 1 νy; λ;,, c ν 1µ l ; λ;,, c ν µ +1 l ] ] t!..3 By settg 1, c e Theorem 1, we ot oe of the results exhted y Zhg d Yg 3, Eq. 8]: Corollry.. For ll tegers µ > 0, ν > 0, α 1, 0 d for λ C, we hve B α νx; λ µ ν +1 S µ 1, λ B α 1 µy; λ 0 0 B α µx; λ ν µ Puttg x 0, y 0 d α 1 Theorem 1, we hve: 0 S ν 1, λ B α 1 νy; λ..4 Corollry.3. For ll tegers µ > 0, ν > 0, 0 d for,, c R + d λ C, we hve the followg relto: ] µ 1ν l B ; λ;,, c µ 1 ν S µ 1, λ l 0 ].5 ν 1µ l B ; λ;,, c ν 1 µ S ν 1, λ l. Flly, susttutg λ 1, 1, c e.5, we fd µ 1 ν B S µ 1 ν 1 µ B S ν result gve y Tueter 18]. 0 Theorem.4. For ll tegers µ > 0, ν > 0, α 1, 0, for,, c R +, d for λ C, we hve the followg detty: λ +j µ ν B α νx + ν l + µν ν µ l ; λ;,, c 0 0 B α µy + jµ l ; λ;,, c ν µ B α µx + µ l + µν ν µ l ; λ;,, c 0 0 λ +j B α νy + jν l ; λ;,, c..7

6 81 Proof. Let the fucto ht e gve y ht tα c µνxt λ µ µνt µνt λ ν µνt µνt λ µt µt α+1 λ νt νt α+1,.8 whch c e expded s follows: ht 1 α µt λ µν α λ µt µt c µνxt µ µνt µνt α νt λ νt νt λ νt νt λ ν µνt µνt λ µt µt α µνt νt µt µt µν α λ µt µt c µνxt λ µ e µνt l α 1 νt λe νt l 1 λ νt νt λ ν e νµt l 1 λe µt l 1 1 α µν α c µν ν µt l µt l c λ µt µt c µνxt λ e νt l 0 α νt λ νt νt λ j e jµt l c µνyt α c µνxt + νt l µν ν µt l + l c l c 1 µν α λ µt λ µt µt 0 α λ j νt λ νt νt + jµt l l c 1 µν α λ B α νx + ν l + µν ν µ l µt ; λ;,, c! 0 λ j B α µy + jµ l νt ; λ;,, c! 0 1 µν α λ +j µ ν B α µy + jµ l ; λ;,, c 0 0 B α νx + ν l ] + µν ν µ l t ; λ;,, c!..9 Sce ht s symmetrc µ d ν, we c lso expd ht s follows: ht 1 λ +j ν µ B α µν α νy + jν l ; λ;,, c 0 0 B α µx + µ l ].10 + µν ν µ l t ; λ;,, c!.

7 8 By equtg the coeffcet of t! o the rght-hd sdes of these lst two.9 d.10, we get the detty.7. Settg 1, c e Theorem yelds result gve recetly y Zhg d Yg 3, Eq. 18] : Corollry.5. For ll tegers µ > 0, ν > 0, α 1, 0 d for λ C, we hve λ +j µ ν B α νx + νµ ; λ B α µy + jµν ; λ λ +j ν µ B α µx + µν ; λ B α νy + jνµ ; λ. Puttg ν 1 d y 0 Theorem gves the ext corollry:.11 Corollry.6. For ll tegers µ > 0, α 1, 0, for,, c R +, d for λ C, we hve the followg detty: λ µ B α λ j µ B α µx + x + l + µ 1 l ; λ;,, c µ 1 l j l ; λ;,, c B α B α 0; λ;,, c ; λ;,, c..1 Theorem.7. For ll tegers µ > 0, ν > 0, α 1, 0, for,, c R +, d for λ C, we hve the followg detty: 0 B α B α λ +j µ ν B α µy; λ;,, c νx + ν l + µν ν µ l + jµ l λ +j ν µ B α νy; λ;,, c µx + µ l + µν ν µ l + jν l ; λ;,, c ; λ;,, c..13 Proof. The proof of Theorem 3 s smlr to tht of Theorems 1 d. I the proof of Theorem 3, we frst me use of 1.7 order to expd the fucto ht defed y.8 d the pply the symmetry of ht µ d ν to ot secod expso of ht. The detls volved re strghtforwrd d we leve them s exercse. If we set 1, c e Theorem 3, we recover result gve recetly y Zhg d Yg 3, Eq. 3] :

8 83 Corollry.8. For ll tegers µ > 0, ν > 0, α 1, 0 d for λ C, we hve λ +j µ ν B α νx + νµ + j; λ B α µy; λ 0 λ +j ν µ B α µx + µ ν + j; λ B α νy; λ. Puttg ν 1 d y 0 Theorem 3 gves the ext corollry:.14 Corollry.9. For ll tegers µ > 0, α 1, 0, for,, c R +, d for λ C, we hve the followg detty: λ µ B α λ j µ B α x + l + µ 1 l ; λ;,, c µx + µ 1 l + j l ; λ;,, c l c B α 0; λ;,, c B α 0; λ;,, c Some dettes relted to geerlzed Apostol-Euler polyomls I ths secto, we derve some dettes cocerg the geerlzed Apostol- Euler polyomls x; λ;,, c defed y 1.8, the geerlzed ltertg sum 1.16 d the logues of the expsos of hyperolc cotget d hyperolc tget troduced 0]. These results exted some ow formuls 6, 7, 1]. We coclude ths secto y gvg some dettes sed o reltoshps etwee the geerlzed Apostol-Euler polyomls x; λ;,, c d the geerlzed Apostol-Beroull polyomls B α x; λ;,, c d etwee the geerlzed Apostol-Beroull polyomls B α x; λ;,, c d the geerlzed Apostol- Geocch polyomls G α x; λ;,, c defed y 1.9. Theorem 3.1. For N 0, µ, ν N, α 1,,, c R +, d for λ C. If µ d ν hve the sme prty, the the followg detty holds true: µ ν l 0 ν µ l 0 νx µx αν l ; λ;,, c T α µ; λ αµ l ; λ;,, c T α ν; λ. 3.1 Proof. Let the fucto gt e gve y gt c µνxt 1 λ 1 µ e µνt l α λ µt + µt α λ νt + νt α 3.

9 84 Mg use of 1.8 d 1.16 to expd gt to ot frst: gt 1 α c ανt l α l c c νxµt 1 λ 1 µ e µνt l λ µt + µt 1 α 1 α 0 E α νx αν l µ ν l µt ; λ;,, c! λe νt l νx α T α µ; λ νt l! αν l ; λ;,, c T α µ; λ t!. 3.3 Now, sce µ d ν hve the sme prty, the the fucto gt s symmetrc µ d ν. Therefore, we c expd gt s follows: gt 1 α ν µ l 0 µx αµ l ; λ;,, c T α ν; λ t!. 3.4 By equtg the coeffcet of t! o the rght-hd sde of the lst two equtos 3.3 d 3.4, we thus recover the detty 3.1 sserted y Theorem 4. As specl cse, f we set 1, c e Theorem 4, we ot the followg corollry gve recetly y Lu d Srvstv 7, Eq. 30]. Corollry 3.. For N 0, µ, ν N, α 1 d for λ C. If µ d ν hve the sme prty, the the followg detty holds true: µ ν α νx ; λ T µ; λ ν µ α µx; λ T ν; λ Now, lettg α λ 1, we recover the result gve y Yg d Qo 1, Eq. 18]: Corollry 3.3. For N 0 d µ, ν N. If µ d ν hve the sme prty, the we hve µ ν E νx T µ ν µ E µx T ν Theorem 3.4. For N 0, µ, ν N, α 1,,, c R +, d for λ C. If µ d ν hve the sme prty, the the followg detty holds true: λ +j µ ν νx + ν + jµ l µ + ν l ; λ;,, c µy; λ;,, c λ +j ν µ 0 µx + µ + jν l µ + ν l ; λ;,, c νy; λ;,, c. 3.7

10 85 Proof. Let the fucto gt e gve y gt c µνxt c 1 µνyt νt µ ] 1 µt ν ] λ λ λ µt + µt α+1 λ νt + νt α whch c e expded, wth the help of 1.8, s follows: gt 1 α c µνxt 1 1 λ νt µ α λ µt + µt νt 1 µt µ+νt α 1 λ µt ν λ µt + 1 λ νt + νt λ νt + 1 α α νt c µνxt λ λ µt + µt 0 jµt α λ j λ νt + νt 1 α λ +j c µνxt + νt l l c α λ µt + µt 0 α λ νt + νt 1 α 1 α λ +j jµt l µ + νt l l c l c νx + ν + µj l µ + ν l ; λ;,, c νt µy; λ;,, c! λ +j µ ν µy; λ;,, c 0 νx + ν + µj l ] µ + ν l ; λ;,, c t!. µt! Usg the fct tht gt s symmetrc sce µ d ν hve the sme prty, we c lso expd gt the followg the wy: gt 1 α λ +j ν µ νy; λ;,, c 0 0 µx + µ + νj l ] µ + ν l ; λ;,, c t!. 3.9 Equtg coeffcets of t! the rght-hd sde of the lst two equtos gves the detty of the Theorem 5. Lettg 1, c e Theorem 5, we fd the followg corollry gve recetly y Lu d Srvstv 7, Eq. 43].

11 86 Corollry 3.5. For N 0, µ, ν N, α 1 d for λ C. If µ d ν hve the sme prty, the the followg detty holds true: λ +j µ ν νx + νµ + j; λ µy; λ λ +j ν µ µx + µ ν + j; λ νy; λ. 0 0 Accordg to 0], we hve the followg logues of the expsos of hyperolc cotget d hyperolc tget, respectvely: λe z + 1 λe z 1 λe z 1 λe z + 1 E 0; λ z! From 3.11, we c ot the followg theorem. B 0; λ + λb 1; λ z 1, 3.11! Theorem 3.6. For N 0, µ, ν N, α 1,,, c R +, d for λ C. Let δ,j deotes the Kroecer delt defed y δ, 1 d δ,j 0 for j. If µ s odd d ν s eve the the followg detty holds true: δ +1,1 + B +1 0; λ] µ µ ν µx µ l ; λ;,, c l j µ ν νx ν l ; λ;,, c l 0 E α 1 µy; λ;,, c. ] + j T 1 j ν; λeα 1 j νy; λ;,, c 0 ] T 1 µ; λ 3.13 Proof. Whe µ s odd d ν s eve, the fucto gt, gve elow, s ot symmetrc µ d ν, so we hve o the oe hd gt 1 α c µνxt 1 λ 1 ν e µνt l 1 λ 1 µ e µνt l α 1 λ νt + νt λ µt + µt 1 λ 1 ν e µνt l α 1 λ µt + µt 1 1 λ 1 µ e µνt l α µt l µνxt α 1 1 λ 1 ν e µνt l c l c λ νt + νt 1 λ 1 ν e µνt l α 1 λe µt l + 1 λ µt + µt 1 1 B 0; λ + λb 1; λ µνt l α 1! µx µ l νt ; λ;,, c! 0

12 µt l T 1 ν ; λ! 0 1 B α 1 0; λ + λb 1; λ] µ 1 0 µ ν 1 µx µ l ; λ;,, c 0 ] ] 1 + j l T 1 j ν; λeα 1 j νy; λ;,, c j E α 1 j νy; λ;,, c µtj j! 87 t ! O the other other hd, we c lso expd gt s follows: gt 1 α c µνxt 1 λ 1 µ e µνt l α 1 λ µt + µt λ νt + νt α 1 λ νt + νt 1 α νt l µνxt 1 λ 1 c l c µ e µνt l α 1 λ µt + µt 1 + λe νt l α 1 λ νt + νt 1 α 1 νx ν l µt νt l ] ; λ;,, c T 1! µ; λ! 0 E α 1 µy; λ;,, c νt! 0 1 µ ν α 1 νx ν l ; λ;,, c 0 0 l ] T 1 µ; λeα 1 µy; λ;,, c Sce λb x + 1; λ B x; λ x 1 see 14] the ] t!. B 0; λ + λb 1; λ δ,1 + B 0; λ Mg use of ths lst relto 3.15 volvg the Apostol-Beroull polyomls d umers d equtg the coeffcets of t! 3.14 d 3.15, we ot Susttutg 1, c e Theorem 6 furshes the followg corollry. Corollry 3.7. For N 0, µ, ν N, α 1 d for λ C. Let δ,j deotes the Kroecer delt defed y δ, 1 d δ,j 0 for j. If µ s odd d ν s eve the the followg detty holds true: δ +1,1 + B +1 0; λ] µ T 1 j ν; λeα 1 j νy; λ j µ ν µx; λ

13 88 µ ν νx; λ T 1 µ; λeα 1 µy; λ Lettg λ , we get the result oted y Lu d Wg 6, Theorem.4]. Corollry 3.8. For N 0, µ, ν N, α 1. If µ s odd d ν s eve the the followg detty holds true: B +1 0 µ µ ν + 1 µx T 1 j j νeα 1 j νy 0 0 µ ν νx T 1 µeα 1 µy We c proceed smlrly to Theorem 3.6, ut usg ths tme 3.1 to estlsh the ext result. 0 0 Theorem 3.9. For 1, µ, ν N, α 1,,, c R +, d for λ C. If µ s eve d ν s odd the the followg detty holds true: E 0; λµ µ ν µx µ l ; λ;,, c 0 0 ] + j l T 1 j ν; λeα 1 j νy; λ;,, c j ν µ µx µ l ; λ;,, c 0 ] 3.18 l T 1 ν; λeα 1 νy; λ;,, c 0 µ ν νx ν l ; λ;,, c 0 ] l T 1 µ; λeα 1 µy; λ;,, c. A terestg specl cse of Theorem 7 s oted y settg 1, c e d λ Corollry For 1, µ, ν N, α 1. If µ s eve d ν s odd the the followg detty holds true: 1 E 0µ µ ν µx T 1 j j νeα 1 j νy 0 0 µ ν νx T 1 µeα 1 µy result gve frst y Lu d Wg 6, Theorem.7].

14 89 Flly, we would le to meto tht my other dettes c e oted from those show ths pper. As exmple, t s esy to see tht the two followg reltoshps hold etwee the Apostol-Beroull d Apostol-Euler polyomls d the Apostol-Beroull d Apostol-Geocch polyomls. Respectvely, we hve for N 0, µ, ν N, α N,,, c R +, d for λ C, d x; λ;,, c α! B α +αx; λ;,, c + α! 3.0 B α x; λ;,, c Gα x; λ;,, c α. 3.1 Let us see two exmples of pplcto of these two results. Frst, comg 3.0 wth Theorem 4 yelds Theorem For N 0, µ, ν N, α N,,, c R +, d for λ C. If µ d ν hve the sme prty, the the followg detty holds true: µ ν! B α αν l +α νx ; λ;,, c l T α µ; λ + α! 0 ν µ! B α +α µx αµ l ; λ;,, c l T α ν; λ. + α! 0 3. Next, cosderg 3.1 wth Theorem 1 gves Theorem 3.1. For ll tegers µ > 0, ν > 0, α N, 0, for,, c R + d for λ C, we hve the followg detty: G α µ 1ν l νx + ; λ;,, c µ ν S µ 1, λ G α 1 µy; λ;,, c l G α µx + ] ν 1µ l ; λ;,, c ν µ +1 S ν 1, λ G α 1 νy; λ;,, c l Refereces ] M. Armowtz d I.A. Stegu, Hdoo of mthemtcl fuctos wth formuls, grphs d mthemtcl tles, Ntol Bureu of Stdrds, Wshgto, DC, Yu. A. Brychov, O multple sums of specl fuctos, Itegrl Trsform Spec. Fuct , L. Comtet, Advced comtorcs: The rt of fte d fte expsos, Trslted from frech y J.W. Nehuys, Redel, Dordrecht, A. Erdely, W. Mgus, F. Oerhettger, d F. Trcom, Hgher trscedetl fuctos, vols.1-3,, E.R. Hse, A tle of seres d products, Pretce-Hll, Eglewood Clffs, NJ, Hogme Lu d Wepg Wg, Some dettes o the Beroull, Euler d Geocch polyomls v power sums d lterte power sums, Dscrete Mthemtcs ,

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