SERIJA III

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1 SERIJA III wwwmahhr/glasni M Riyasa and S Khan Some resuls on -Hermie based hybrid polynomials Acceped manuscrip This is a preliminary PDF of he auhor-produced manuscrip ha has been peer-reviewed and acceped for publicaion I has no been copyedied, proofread, or finalized by Glasni Producion saff

2 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS Mumaz Riyasa and Subuhi Khan Deparmen of Mahemaics, Faculy of Science, Aligarh Muslim Universiy, Aligarh, India Absrac In his aricle, a hybrid class of he -Hermie based Aposol ype Frobenius-Euler polynomials is inroduced by means of generaing funcion and series represenaion Several imporan formulas and recurrence relaions for hese polynomials are derived via dieren generaing funcion mehods Furher, he D -Hermie based Aposol-Bernoulli, Aposol-Euler and Aposol-Genocchi polynomials are inroduced and imporan relaions for hese polynomials are also esablished Finally, a new class of he D -Hermie based Appell polynomials is invesigaed as he generalizaion of he above polynomials The deerminan deniions for he D -Hermie based Appell and relaed polynomials are also explored Inroducion and preliminaries The subjec of -calculus sared appearing in he nineeenh cenury due o is applicaions in various elds of mahemaics, physics and engineering The deniions and noaions of -calculus reviewed here are aen from The -analogue of he shifed facorial a n is given by n a; 0 =, a; n = m a, n N m=0 The -analogues of a complex number a and of he facorial funcion are given by a = a, C {}; a C, 000 Mahemaics Subjec Classicaion B73, B83, B68 Key words and phrases -Hermie ype polynomials, Aposol ype -Frobenius-Euler polynomials, -Hermie based Aposol ype Frobenius-Euler polynomials

3 M RIYASAT AND S KHAN 3 n! = 4 n m = n = ; n, ; n N, n m= 0! =, C; 0 < < is given by The Gauss -binomial coecien n = n!!n! = =0 ; n ; ; n, = 0,,, n The -analogue of he funcion x + y n is given by n 5 x + y n := / x n y, n N 0 The -analogues of exponenial funcions are given by x n 6 e x = n! :=, 0 < < ; x <, x; 7 E x = nn / xn n! := x;, 0 < < ; x C Moreover, he funcions e x and E x saisfy he following properies: 8 D e x = e x, D E x = E x, where he -derivaive D f of a funcion f a a poin 0 z C is dened as follows: 9 D fz = fz fz, 0 < < z z For any wo arbirary funcions fz and gz, he -derivaive operaor D saises he following produc and uoien relaions: 0 D,z fzgz = fzd,z gz + gzd,z fz, fz D,z gz = gzd,zfz fzd,z gz gzgz Recenly, exensive invesigaions relaed o he -Bernoulli polynomials B n, x, -Euler polynomials E n, x and -Genocchi polynomials G n, x and heir generalizaions in wo variables x and y are considered, see for example 6, 8, 9, 0, 7, 0,, 5, 4 We recall he following deniions:

4 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 3 Definiion The -Aposol-Bernoulli polynomials B α n,x, y; λ of order α C, α N, 0 < < in x and y are dened by 9 λe α e xe y = B n,x, α y; λ n, + log λ < π, n! where B α n,λ := B α n,0, 0; λ are he -Aposol-Bernoulli numbers Definiion The -Aposol-Euler polynomials E α n,x, y; λ of order α C, α N, 0 < < in x and y are dened by 9 3 λe + α e xe y = E α n,x, y; λ n, + log λ < π, n! where E α n,λ := E α n,0, 0; λ are he -Aposol-Euler numbers Definiion 3 The -Aposol-Genocchi polynomials G α n, x, y; λ of order α C, α N, 0 < < in x and y are dened by 9 4 λe + α e xe y = G n, α x, y; λ n, + log λ < π, n! where G α n, λ := G α n, 0, 0; λ are he -Aposol-Genocchi numbers Several unied forms of he Aposol-ype polynomials are inroduced and sudied by many auhors, for his see 7,, 3,, 7, 8, 9,, 5 We recall he deniion of he Aposol ype -Frobenius-Euler polynomials H α n,x, y; u; λ inroduced and sudied by Bura Kur in 3: Definiion 4 The Aposol ype -Frobenius-Euler polynomials H n,x, α y; u; λ of order α C, α N, 0 < < in x and y are dened by α u 5 e xe y = H α λe u n,x, y; u; λ n n!, where H n,u; α λ := H n,0, α 0; u; λ are he Aposol ype -Frobenius-Euler numbers dened by α u 6 = H α λe u n,0, 0; u; λ n n! For y = 0, he polynomials H n,x, α y, u; λ reduce o he -Aposol ype Frobenius-Euler polynomials H n,x, α u; λ in one variable 6

5 4 M RIYASAT AND S KHAN By leing and λ =, he polynomials H n,x, α u; λ reduce o he Frobenius-Euler polynomials H n α x, u 4, 3 Very recenly a new ype of -Hermie polynomial is considered in 4, which is a paricular member of he -Appell family The -Appell polynomials are dened by means of he following generaing funcion: 7 g e x = A n, x n n!, A n, := A n, 0 Definiion 5 The coninuous -Hermie polynomials H n,x s 0 < <, 0 s R are dened by 8 e x s + = H s n,x n n!, where H n, s := H n,0 s are he -Hermie numbers dened by 9 e s = H s + n,0 n n! The coninuous -Hermie polynomials H n,x s are -Appell for g = s e + To sudy hybrid forms of he -polynomials by dieren means is a new approach Very recenly, Riyasa e al 4 inroduced and sudied he composie D -Appell polynomials In order o exend his approach, in his aricle, a hybrid class of he -Hermie based Aposol ype Frobenius-Euler polynomials is inroduced The generaing funcion, series represenaion and several imporan formulas and relaions for hese polynomials are derived The D -Hermie based Aposol-Bernoulli, Aposol-Euler and Aposol-Genocchi polynomials are also inroduced and corresponding resuls are esablished Finally, a new family of he D -Hermie based Appell polynomials is inroduced and sudied from deerminan poin of view -Hermie based Aposol ype Frobenius-Euler polynomials In his secion, we inroduce he -Hermie based Aposol ype Frobenius- Euler polynomials HbATFEP by means of generaing funcion and series represenaion Cerain relaions for hese polynomials are also derived by using various ideniies In order o esablish he generaing funcion for he HbATFEP, he following resul is proved:

6 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 5 Theorem Le C, α N, 0 < < The following generaing funcion for he -Hermie based Aposol ype Frobenius-Euler polynomials HH n, α,s x, y; u; λ of order α holds rue: 0 α u e x s E y = HH n, α,s x, y; u; λ n λe u + n! Proof Expanding he exponenial funcion e x and hen replacing he powers of x, ie x 0, x, x,, x n by he corresponding polynomials H s 0, x, Hs, x,, Hs n,x in he lhs and replacing x by he polynomial H s, x in he rhs of euaion 5 and afer summing up he erms in he lhs of he resulan euaion, we nd α u E y H s λe u n,x n n! = H n,h α s n, x, y; u; λ n!, which on using euaion 8 in he lhs and denoing he resulan HbATFEP in he rhs by H H α,s n, x, y; u; λ yields asserion 0 Noe We noe ha H H n, α,s based Aposol ype Frobenius-Euler numbers dened by u λe u u; λ := H H n, α,s 0, 0; u; λ are he -Hermie α e s + = HH n, α,s u; λ n n! Theorem The following series represenaion for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ of order α hold rue: n 3 HH n, α,s x, y; u; λ = H α n, 0, y; u; λhs, x =0 Proof Using euaions 5 and 8 in he lhs of euaion 0 and hen applying he Cauchy produc rule and euaing he coeciens of same powers of in boh sides of resulan euaion, we ge represenaion 3 Theorem 3 The following summaion formulas for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ of order α holds rue: n 4 HH n, α,s x, y; u; λ = HH α, 0, 0; u; λx + yn, =0

7 6 M RIYASAT AND S KHAN 5 HH α,s n, x, y; u; λ = =0 n HH α, 0, y; u; λxn, 6 HH α,s n, x, y; u; λ = =0 n n n / HH α, x, 0; u; λyn Proof Suiably using euaions 5-7 in generaing funcion 0 o ge hree dieren forms Furher, maing use of he Cauchy produc rule in he resulan expressions and hen comparing he lie powers of in boh sides of resulan euaion, we nd formulas 4-6 Theorem 4 The following recursive formulas for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ of order α hold rue: 7 D,x H H n, α,s x, y; u; λ = n H H α,s n, x, y; u; λ, 8 D,y H H n, α,s x, y; u; λ = n H H α,s n, x, y; u; λ Proof Diereniaing generaing funcion 0 wih respec o x and y wih he help of euaion 8 and hen simplifying wih he help of he Cauchy produc rule, formulas 7 and 8 are obained Theorem 5 The following recurrence relaion for he -Hermie based Aposol ype Frobenius-Euler polynomials H H s n,x, y; u; λ holds rue: 9 HH s n+, x, y; u; λ = s + + y H H n,x, s y; u; λ λ u n H H s n, x, y; u; λ + x HH s =0 n,x, y; u; λ n HH s n, x, y; u; λn H,, 0; u; λ Proof Taing α = and hen applying -derivaive on boh sides of generaing funcion 0, i follows ha 30 HH s n+, n x, y; u; λ n! = ud e xe ye, λe u s +,

8 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 7 which on performing diereniaion in he lhs using formula yields 3 HH s n n+, x, y; u; λ = u λe ud, e xe ye n! λe uλe u e xe ye s D +,λe u λe uλe u s + Now, using produc diereniaion formula 0 in he above euaion, i follows ha 3 HH s n n+, x, y; u; λ n = s u! + λe u e x s + E y +x u λe u λ u e x s + E y + y e x s u λe u + u λe u E y e x s + e u λe u E y Furher, using generaing funcions 5 and 0 wih α = in euaion 3, we nd 33 HH s n n+, x, y; u; λ +x HH s HH s λ u n! = n s + HH s n,x, y; u; λ n n + y! n,x, y; u; λ n n n! =0 HH s n n, x, y; u; λ n! n,x, y; u; λ n H,, 0; u; λ!, which on maing use of he Cauchy produc rule in he rhs and comparing he coeciens of n /n! on boh sides of he resulan euaion gives recurrence relaion 9 Theorem 6 The following relaion for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n,x, s y; u; λ holds rue: n 34 u HH s n, x, y; u; λhs, 0, 0; u; λ =0 n! = u H H s n,x, y; u; λ u H H s n,x, y; u; λ Proof Maing use of he ideniy 35 u λe uλe u = λe u λe u

9 8 M RIYASAT AND S KHAN o evaluae he following fracion, so ha we have ue x s + ue y u λe uλe u 36 ue x s + ue y = λe u ue x s + ue y, λe u which on using euaions 0 and 6 in boh sides gives 37 u HH n,x, s y; u; λ n n! H s, 0, 0; u; λ! = u =0 HH s n,x, y; u; λ n n! u HH s n,x, y; u; λ n n! Applying he Cauchy produc rule in he above euaion and hen euaing he coeciens of lie powers of in boh sides of he resulan euaion, asserion 34 follows Theorem 7 The following relaion for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n,x, s y; u; λ holds rue: 38 n u H H n,x, s y; u; λ = λ H H s n, x, y; u; λ uhs n, 0x + y 39 =0 Proof By using e E =, we consider he following ideniy: u λλe ue = λe u λe Evaluaing he following fracion using above ideniy, we nd u ue x s + E y ue x s + E y 40 = λλe ue λe u ue x s + E y, λe which on using euaions 0, 5 and 9 yields 4 u HH n,x, s y; u; λ n n = λ! HH n,x, s y; u; λ n u H n,0 s n n! x + y =0 n! =0!!

10 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 9 Maing use of he Cauchy produc rule in he rhs of above euaion and hen comparing he coeciens of n /n! on boh sides of he resulan euaion, we ge relaion 38 Theorem 8 The following relaion for he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ of order α holds rue: 4 HH α,s n, x, y; u; λ = u 43 =0 n λ H H α,s n,, y; u; λh,x, 0; u; λ u H H α,s n, x, 0; u; λh,0, y; u; λ Proof Consider generaing funcion 0 in he following form: HH n, α,s x, y; u; λ n n! = u E y λe u e x s + λe u u u α λe u Simplifying he above euaion and using euaions 0 and 5, we nd 44 =0 HH n, α,s x, y; u; λ n n =! u λ H, x, 0; u; λ! u HH n, α,s HH n, α,s x, 0; u; λ n n!, y; u; λ n n! =0 H, 0, y; u; λ! Applicaion of he Cauchy produc rule in he rhs and cancellaion of he coeciens of same powers of in boh sides of he resulan euaion yields relaion 4 Theorem 9 The following relaion beween he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ and he -Aposol- Bernoulli polynomials B n, x, y; λ holds rue: 45 n+ n + n, x, y; u; λ = n + =0 B, 0, y; λ HH α,s λ B r, 0, y; λ r HH α,s n +, x, 0; u; λ

11 0 M RIYASAT AND S KHAN 46 Proof Consider generaing funcion 0 in he following form: HH n, α,s x, y; u; λ n n =! α u E y λe u e x s, + λe λe which on simplicaion and use of euaions 0 and wih α = gives 47 HH n, α,s x, y; u; λ n n =! λ! r r! HH n, α,s x, 0; u; λ n n! HH n, α,s x, 0; u; λ n n! =0 B, 0, y; λ =0 B, 0, y; λ! On euaing he coeciens of same powers of afer using Cauchy produc rule in euaion 47, asserion 45 follows Theorem 0 The following relaion beween he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ and he -Aposol- Euler polynomials E n, x, y; λ holds rue: HH n, α,s x, y; u; λ = n 48 λ E r, x, 0; λ =0 r + E, x, 0; λ 0, y; u; λ 49 HH α,s n, Proof Consider generaing funcion 0 in he following form: HH n, α,s x, y; u; λ n n =! α u E y λe u e x s + λe + λe + Simplifying he above euaion and hen maing use of euaions 0 and 3 wih α =, i follows ha 50 HH n, α,s x, y; u; λ n n =! λ r! r +! HH n, α,s 0, y; u; λ n n! HH n, α,s 0, y; u; λ n n! =0 E, x, 0; λ! =0 E, x, 0; λ, which on using he Cauchy produc rule and euaing he coeciens of same powers of in resulan euaion yields relaion 48

12 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS Theorem The following relaion beween he -Hermie based Aposol ype Frobenius-Euler polynomials H H n, α,s x, y; u; λ and he -Aposol- Genocchi polynomials G n, x, y; λ holds rue: 5 5 HH n, α,s x, y; u; λ = n+ =0 n + n + + G, 0, y; λ λ HH α,s n +, G r, 0, y; λ r x, 0; u; λ Proof Consider generaing funcion 0 in he following form: HH n, α,s x, y; u; λ n n =! α u E y λe u e x s + λe + λe + Simplifying euaion 5 and using euaions 0 and 4 wih α =, we nd 53 HH n, α,s x, y; u; λ n n =! λ! r r +! HH n, α,s x, 0; u; λ n n! HH n, α,s x, 0; u; λ n n! =0 G, 0, y; λ =0 G, 0, y; λ! Comparison of he lie powers of n /n! afer using he Cauchy produc rule in he above euaion yields desired ideniy 5 In he nex secion, we inroduce he D -Hermie based Aposol-Bernoulli, Aposol-Euler and Aposol-Genocchi polynomials and esablish some relaions for hese hybrid polynomials 3 D -Hermie based Aposol-Bernoulli, Aposol-Euler and Aposol-Genocchi polynomials Kelesheri and Mahmudov in 0 inroduced and sudied he D -Appell polynomials A n, x, y which are he -variable generalizaions of he -Appell polynomials A n, x The D -Appell polynomials A n, x, y are dened by means of he following generaing funcion: 354 g e xe y = A n, x, y n n!, A n, := A n, 0, 0 The D -Bernoulli polynomials B n, x, y, -Euler polynomials E n, x, y and -Genocchi polynomials G n, x, y are he paricular members of he D

13 M RIYASAT AND S KHAN -Appell family Several imporan relaions and formulas for hese polynomials and for heir generalizaions are derived in 6, 8, 0, 5, 6 The approach used in previous secion is furher exploied o inroduce he D -Hermie-based Aposol-Bernoulli, Aposol-Euler and Aposol-Genocchi polynomials Firs, we give he following deniions: Definiion 3 The D -Hermie based Aposol-Bernoulli polynomials DHbABP of order α, H B α,s n, x, y; u; λ C, α N, 0 < < are dened by he following generaing funcion: α 355 e x s E y = HB α,s n, x, y; λ n λe + n!, where H B α,s n, Bernoulli numbers g = λe s + λ := H B α,s n, 0, 0; λ are he D -Hermie-based Aposol- The DHbABP H B α,s n, x, y; λ are D -Appell for α e Definiion 3 The D -Hermie based Aposol-Euler polynomials HE α,s n, x, y; u; λ DHbAEP of order α C, α N, 0 < < are dened by he following generaing funcion: α 356 e x s E y = HE α,s n, x, y; λ n λe + + n!, where H E α,s n, Euler numbers The DHbAEP H E α,s λe+ α e λ := H E α,s n, 0, 0; λ are he D -Hermie-based Aposoln, x, y; λ are D -Appell for g = s + Definiion 33 The D -Hermie based Aposol-Genocchi polynomials HG n, α,s x, y; u; λ DHbAGP of order α C, α N, 0 < < are dened by he following generaing funcion: α 357 e x s E y = HG n, α,s x, y; λ n λe + + n!, where H G n, α,s Genocchi numbers g = λe+ s + λ := H G n, α,s 0, 0; λ are he D -Hermie-based Aposol- The DHbAGP H G n, α,s x, y; λ are D -Appell for α e Analogous o he resuls obained for he HbATFEP in Secion, we obain he series represenaions, summaion formulas and recursive formulas for he DHbABP, DHbAEP and DHbAGP We presen hese resuls

14 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 3 in TABLE I, II-IV, V-VI, respecively Table Resuls for H B α,s n, x, y; λ, H E α,s n, x, y; λ, H G n, α,s x, y; λ S Resuls for H B α,s n, x, y; λ Resuls for H E n, α,s x, y; λ Resuls for H G n, α,s x, y; λ No I H B α,s n, x, y; λ = n =0 H s, x Bα 0, y; λ n, H E α,s n, x, y; λ = n =0 Hs, x Eα 0, y; λ n, H G α,s n, x, y; λ = n =0 Hs, x Gα 0, y; λ n, II H B α,s n, x, y; λ = n H B α, λ =0 x + y n H E α,s n, x, y; λ = n H E α, λ =0 x + y n H G α,s n, x, y; λ = n H G α, λ =0 x + y n III IV H B α,s n, x, y; λ = n H B α,s n, x, y; λ = n n n H E α,s n, x, y; λ = n =0 H B α, x, 0; λyn n n H G α,s n, x, y; λ = n =0 H E α, x, 0; λyn =0 H B α 0, y; λ, H E α,s n, x, y; λ = n =0 H E α 0, y; λ, H G n, α,s x, y; λ = n =0 x n x n x n n n =0 H G α x, 0; λyn, H G α, 0, y; λ V D,x H B α,s n, x, y; λ D,x H E α,s n, x, y; λ D,x H G α,s = n H B α,s n, x, y; λ = n H E α,s n, x, y; λ = n, x, y; λ n H G α,s n, x, y; λ VI D,y H B α,s n, x, y; λ D,y H E α,s n, x, y; λ D,y H G α,s = n H B α,s n, x, y; λ = n H E α,s n, x, y; λ = n, x, y; λ n H G α,s n, x, y; λ Nex, we esablish cerain summaion relaions for he DHbABP, DHbAEP and DHbAGP by proving he following resuls: Theorem 34 The following relaions for he D -Hermie based Aposol- Bernoulli polynomials H B α,s n, x, y; λ hold rue: 358 a H B α,s n, x, y; λ = m n +m λ HB m α,s, x, 0; λ E n, 0, my; λ, =0 r mr HB α,s r, x, 0; λ 359 b H B α,s n, x, y; λ = m n +m λ HB m α,s, 0, y; λ E n, mx, 0; λ, =0 r mr HB α,s r, 0, y; λ

15 4 M RIYASAT AND S KHAN 360 c H B α,s n, x, y; λ = n+ m n HB α,s r, n+ n+ =0 0, y; λ + H B α,s, 0, y; λ 36 d H B α,s n, x, y; λ = n+ m n HB α,s r, n+ n+ =0 x, 0; λ + H B α,s, x, 0; λ m m λ r G n +, mx, 0; λ, m m λ r G n +, 0, my; λ mr mr Proof a Consider generaing funcion 355 in he following form: 36 α HB α,s n, x, y; λ n n =! λe e x s + E m my λe/m+ λe /m+, which on simplicaion becomes 363 HB α,s n, λe /m+ x, y; λ n e m n! = + λ λe λe α e x s + E m my E m my λe /m+ α e x s Now, using euaions 355 and 3 in euaion 363, we nd HB α,s n, x, y; λ n n =! λ m n E n, 0, my; λ n n! r! r +! m n E n, 0, my; λ n n! HB α,s r, x, 0; λ m =0 =0 HB α,s, x, 0; λ! Furher, maing use of he Cauchy produc rule in he rhs of above euaion and comparing he coeciens of lie powers of, asserion 358 follows b Consider generaing funcion 355 in he following form: 365 HB α,s n, x, y; λ n n! = λe α e m mx e λe /m+ s + λe/m+ E y Following he same lines of proof as in a, we are led o asserion 359

16 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 5 c Consider generaing funcion 355 in he following form: 366 HB α,s n, x, y; λ n n! = λe α e m mx e λe /m+ s + λe/m+ E y Simplifying he above euaion and using euaions 355 and 4, we nd 367 HB α,s n, x, y; λ n n =! m n λ HB α,s r, 0, y; λ r! r +! m n G n, mx, 0; λ n n! G n, mx, 0; λ n n! m =0 =0 HB α,s, 0, y; λ! Use of he Cauchy produc rule in he rhs of he simplied form of he above euaion and nally euaing he coeciens of same powers of gives asserion 360 d Consider generaing funcion 355 in he following form: 368 α HB α,s n, x, y; λ n n =! λe E m my e x s + λe/m+ λe /m+ Proceeding on he same lines of proof as in c, we are led o relaion 36 Theorem 35 The following relaions for he D -Hermie based Aposol- Euler polynomials H E α,s n, x, y; λ hold rue: 369 a H E α,s n, x, y; λ = n+ n+ m n n+ m m λ r mr =0 HE α,s r, 0, y; λ H E α,s, 0, y; λ B n +, mx, 0; λ, 370 b H E α,s n, x, y; λ = n+ m n HE α,s r, n+ n+ =0 x, 0; λ H E α,s, x, 0; λ 37 c H E α,s n, x, y; λ = n+ m n HE α,s r, n+ n+ =0 0, y; λ + H E α,s, 0, y; λ m m λ r B n +, 0, my; λ m m λ r G n +, mx, 0; λ, mr mr

17 6 M RIYASAT AND S KHAN 37 d H E α,s n, x, y; λ = n+ m n HE α,s r, n+ n+ =0 x, 0; λ + H E α,s, x, 0; λ m m λ r G n +, 0, my; λ Proof Taing suiable arrangemens of generaing funcion 356 and proceeding on he same lines of proof as in Theorem 34, asserions can be proved Thus, we omi i Theorem 36 The following relaions for he D -Hermie based Aposol- Genocchi polynomials H G n, α,s x, y; λ hold rue: 373 a H G n, α,s x, y; λ = n+ n+ m n n+ m m λ r mr =0 HG r, α,s 0, y; λ H G α,s, 0, y; λ B n +, mx, 0; λ, 374 b H G n, α,s x, y; λ = n+ m n HG r, α,s c H G n, α,s x, y; λ = m n n+ n+ =0 x, 0; λ H G α,s, x, 0; λ =0 r mr HG α,s r, x, 0; λ d H G n, α,s x, y; λ = m n =0 r mr HG α,s r, 0, y; λ m m λ r B n +, 0, my; λ mr mr HG m α,s, x, 0; λ + m λ E n, 0, my; λ, HG m α,s, 0, y; λ + m λ E n, mx, 0; λ Proof Considering appropriae arrangemens of generaing funcion 34 and following he same lines of proof as in Theorem 34, we ge asserions Thus, we omi i In he nex secion, we inroduce a new class of he D -Hermie based Appell polynomials DHbAP by means of generaing funcion and series represenaion 4 D -Hermie based Appell polynomials Firs, we esablish he generaing funcion for he DHbAP by maing use of replacemen echniue For, his we consider he following deniions:

18 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 7 Definiion 4 The D -Hermie based Appell polynomials H A s n,x, y C, 0 < < are dened by means of he following generaing funcion: 477 g e x s E y = + HA s n,x, y n n!, HA s n, := H A s n,0, 0 Definiion 4 The D -Hermie based Appell polynomials H A s n,x, y are dened by he following series represenaion: 478 HA s n,x, y = =0 n A, x, yh s n, 0 Noe We noe ha by aing g = in euaion 4, he DHbAP HA s n,x, y reduce o D -Hermie polynomials H n,x, s y Thus, we have he following deniion: Definiion 43 The D -Hermie polynomials H s n,x, y are dened by means of he following generaing funcion: 479 e x s E y = + H s n,x, y n n! Remar 44 From deniions 4 and 43, i is clear ha 480 HA s n,x, 0 = H A s n,x, H s n,x, 0 = H s n,x, where H A s n,x and H n,x s are he -Hermie based Appell polynomials and he -Hermie polynomials, respecively To sudy he wo-variable forms of he -polynomials from deerminan poin of view is a new invesigaion In paricular, he deerminan deniion for he D -Appell polynomials is considered in 0 Moivaed by his, we nd he deerminan deniion for he DHbAP H A s n,x, y By using a similar approach as in 0, p359 Theorem 7 and aing help of euaions 477 and 479, we nd he following deerminan deniion for he DHbAP H A s n,x, y:

19 8 M RIYASAT AND S KHAN Definiion 45 The D -Hermie based Appell polynomials H A s n,x, y of degree n are dened by 48 HA s 0, x, y = β 0,, HA s n,x, y = n β 0, n+ H s, x, y Hs, x, y Hs n, x, y Hs n,x, y β 0, β, β, β n, β n, 0 β 0, β, βn, βn,, 0 0 β 0, n βn 3, βn, β n 0, n β, where n =,,, β 0, 0; β 0,, β,, β,,, β n, R For suiable choices of g, dieren members belonging o he family of DHbAP can be obained Paricularly, we noe ha he DHbABP, DHbAEP and DHbAGP are he special members of he DHbAP H A s n,x, y We conclude ha for λ =, he polynomials H B α,s n, x, y; λ, H E n, α,s x, y; λ and H G n, α,s x, y; λ reduce o he D -Hermie based Bernoulli polynomials HB n, α,s x, y, D -Hermie based Euler polynomials H E n, α,s x, y and D -Hermie based Genocchi polynomials H G α,s n, x, y, each of order α For α =, hese polynomials reduce o he D -Hermie based Bernoulli polynomials H B n,x, s y, D -Hermie based Euler polynomials H E n,x, s y and D -Hermie based Genocchi polynomials H G s n,x, y Recenly, Riyasa e al in 5 gave he deerminan deniions of he -Bernoulli, -Euler and -Genocchi polynomials Firs, we slighly focus on he deerminan deniions of he D -Bernoulli, -Euler and -Genocchi polynomials By aing β 0, =, β i, = i+ ; β0, =, β i, = and β0, =, β i, = i+ i =,,, n, respecively in deerminan deniion of he D -Appell polynomials 0, p359 Theorem 7, we can obain he deerminan deniions of he D -Bernoulli, -Euler and -Genocchi polynomials B n, x, y, E n, x, y and G n, x, y, respecively

20 SOME RESULTS ON -HERMITE BASED HYBRID POLYNOMIALS 9 Again, by aing β 0, =, β i, = i+ ; β0, =, β i, = and β0, =, β i, = i+ i =,,, n, respecively in deerminan deniion 48 of he D -Hermie based Appell polynomials, we nd he following deerminan deniions for he polynomials H B n,x, s y, H E n,x, s y and HG s n,x, y, respecively: Definiion 46 The D -Hermie based Bernoulli polynomials H B s n,x, y of degree n are dened by 48 HB s 0, x, y =, HB n,x, s y = n H s, x, y Hs, 0 x, y Hs 3 n, x, y Hs n,x, y n n+ n n 0 0 n n n n n, n =,, Definiion 47 The D -Hermie based Euler polynomials H E s n,x, y of degree n are dened by 483 HE s 0, x, y =, HE n,x, s y = n H s, x, y Hs, 0 x, y Hs n, n x, y Hs n,x, y 0 0 n n n, n =,,

21 0 M RIYASAT AND S KHAN Definiion 48 The D -Hermie based Genocchi polynomials H G s n,x, y of degree n are dened by 484 HG s 0, x, y =, HG s n,x, y = n H s, x, y Hs, 0 x, y Hs 3 n, x, y Hs n,x, y n n+ n n 0 0 n n n n n, n =,, The -dierence euaions for he D -Appell and composie D -Appell polynomials are esablished in, 4 This provides moivaion o esablish -dierence euaions for he D -Hermie based Appell polynomials and also for heir composie forms This aspec will be aen in nex invesigaion Acnowledgemens This wor has been done under Pos-Docoral Fellowship Oce Memo No/4038/06/R&D-II/063 awarded o Dr Mumaz Riyasa by he Naional Board of Higher Mahemaics, Deparmen of Aomic Energy, Governmen of India, Mumbai The auhors are hanful o he reviewers for several useful commens and suggesions owards he improvemen of his paper References WA Al-Salam, -Appell polynomials, Ann Ma Pura Appl , 345 GE Andrews, R Asey and R Roy, 7h Special funcions of Encyclopedia of Mahemaics and is Applicaions, Cambridge Universiy Press, Cambridge, L Carliz, Eulerian numbers and polynomials, Mah Mag 3 959, Gi-Sang Cheon and Ji-Hwan Jung, The -Sheer seuences of a new ype and associaed orhogonal polynomials, 49 06, J Choi, PJ Anderson and HM Srivasava, Some -exensions of he Aposol- Bernoulli and Aposol-Euler polynomials of order n and he muliple Hurwiz zea funcions, Appl Mah Compu , J Choi, PJ Anderson and HM Srivasava, Carliz's -Bernoulli and -Euler numbers and polynomials and a class of generalized -Hurwiz zea funcions, Appl Mah Compu , 8508

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23 M RIYASAT AND S KHAN Mumaz Riyasa Deparmen of Mahemaics Faculy of Science Aligarh Muslim Universiy Aligarh India mumazrs@gmailcom Subuhi Khan Deparmen of Mahemaics Faculy of Science Aligarh Muslim Universiy Aligarh India subuhi006@gmailcom