Some new classes of generalized Lagrange-based Apostol type Hermite polynomials

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1 Preprins NOT PEER-REVIEWED Posed: 19 Juy 2018 Some new casses of generaized Lagrange-based Aposo ype Hermie poynomias Waseem A. Khan 1 and Nisar K S 2 1 Deparmen of Mahemaics, Facuy of Science, Inegra Universiy, Lucknow , India 2 Deparmen of Mahemaics, Coege of Ars and Science-Wadi A dawaser, Prince Saam bin Abduaziz Universiy, Riyadh region 11991, Saudi Arabia E-mai: waseem08 khan@rediffmai.com, n.sooppy@psau.edu.sa Absrac. In his paper, we inroduce a genera famiy of Lagrange-based Aposoype Hermie poynomias hereby unifying he Lagrange-based Aposo Hermie- Bernoui and he Lagrange-based Aposo Hermie-Genocchi poynomias. We aso define Lagrange-based Aposo Hermie-Euer poynomias via he generaing funcion. In erms of hese generaizaions, we find new and usefu reaions beween he unified famiy and he Aposo Hermie-Euer poynomias. We aso derive heir expici represenaions and is some basic properies of each of hem. Some impici summaion formuae and genera symmery ideniies are derived by using differen anayica means and appying generaing funcions Mahemaics Subec Cassificaion: 11B68, 11B65, 33C45. Keywords: Hermie poynomias, Chan-Chyan-Srivasava poynomias, Lagrangebased Aposo ype Hermie poynomias, summaion formuae, symmeric ideniies. 1. Inroducion The Lagrange poynomias in severa variabes, which are known as he Chan- Chyan-Srivasava poynomias [3] are defined by means of he foowing generaing funcion: 1 x α 1 g α 1...α r n x 1,..., x r n 1.1 α C 1,..., r; < min{ x 1 1,..., x r 1 } and are given expiciy by g n α1,α2...αr x 1, x 2,..., x r... k r 10 xk 1 1 k 1! x k 2 k 1 2 k 3 k 2 k 2 α1 k1 α 2 k2 k 1...α r 1 kr 1 k r 2α r n kr 1 x n k r 1 r k 2 k 1!... x kr 1 kr 2 r 1 k r 1 k r 2! n k r 1!, 1.2 where and in wha foows λ n denoes he Pochhamer symbo or he shifed facoria defined by λ 0 : 1 and λ n λλ...λ + n 1n N : {1, 2, 3,...}. Luo and Srivasava [10, 11, 12] inroduced he generaized Aposo-Bernoui poynomias B n α x of order α, Luo [7] invesigaed he generaized Aposo-Euer poynomias E n α x of order α and he generaized Aposo-Genocchi poynomias G α n x of order α see aso [6, 8, 9]. The generaized Aposo-Bernoui poynomias B n α x; λ of order α C, he generaized Aposo-Euer poynomias E n α x; λ of order α C, he generaized 2018 by he auhors. Disribued under a Creaive Commons CC BY icense.

2 Preprins NOT PEER-REVIEWED Posed: 19 Juy Aposo-Genocchi poynomias G α n x; λ of order α C are defined respecivey by he foowing generaing funcions: and λe 1 I is easy o see ha B α n α e x α 2 λe e x 1 α 2 λe e x 1 x B α x; 1, E α x E α n n B n α x; λ n, < n λ < 2π; 1α E n α x; λ n, < n λ < π; 1α G α n x; λ n, < n λ < π; 1α n x; 1 and G α n x G α n x; 1. Receny, Srivasava e a. [16] inroduce and invesigae he foowing cass of Lagrange-based Aposo ype poynomias T α 1,,α r ;α n.λ,k x 1 x r ; x as foows: Definiion 1.1. The Lagrange-based Aposo ype poynomias T α 1,,α r ;α,k x 1 x r ; x are defined by means of he foowing generaing funcion: 2 k α 1 x α λe + 1 k+1 e x T α1,,αr;α,k x 1 x r ; x n, λ C; α C 1,, r. In he paricuar cases when k 0 and k 1, we define he Lagrange-based Aposo-Bernoui poynomias B α 1,,α r ;α x 1 x r ; x and he Lagrange-based Aposo- Genocchi poynomias G α 1,,α r ;α x 1 x r ; x as foows. Definiion 1.2. The Lagrange-based Aposo-Bernoui poynomias B α1,,αr;α x 1 x r ; x given by 1 B α 1,,α r ;α x 1 x r ; x T α 1,,α r ;α,0 x 1 x r ; x are defined by means of he foowing generaing funcion α 1 x α λe e x B α 1,,α r ;α x 1 x r ; x n, λ C; α C 1,, r. Definiion 1.3. The Lagrange-based Aposo-Genocchi poynomias G α 1,,α r ;α x 1 x r ; x given by 1 G α 1,,α r ;α x 1 x r ; x T α 1,,α r ;α,1 x 1 x r ; x are defined by means of he foowing generaing funcion α 1 x α 2 λe e x G α 1,,α r ;α x 1 x r ; x n, λ C; α C 1,, r. Furhermore, we define he Lagrange-based Aposo-Euer poynomias as foows.

3 Preprins NOT PEER-REVIEWED Posed: 19 Juy 2018 Definiion 1.4. The Lagrange-based Aposo-Euer poynomias E α 1,,α r ;α x 1 x r ; x are defined by means of he foowing generaing funcion: 1 x α 1 2 λe + 1 α e x λ C; α C 1,, r. E α 1,,α r ;α x 1 x r ; x n, 1.9 The 2-variabe Hermie Kampé de Férie poynomias 2VHKdFP H n x, y [2, 4] are defined as e x+y2 H n x, y n, 1.10 are souions of he hea equaion y H nx, y 2 x 2 H nx, y, 1.11 H n x, 0 x n. The higher order Hermie poynomias, someimes caed he Kampe de Ferie poynomias of order m or he Goud Hopper poynomias GHP H n m x, y defined by he generaing funcion [1] e x+ym are souions of he generaized hea equaion [4] Aso we noe ha H n m x, y n where H n x are he cassica Hermie poynomias [1] m fx, y fx, y 1.13 y xm fx, 0 x n H 2 n x, y H n x, y, 1.15 H n 2x, 1 H n x, 1.16 The Lagrange-based generaizaions which we have inroduced above, enabe us o obain new and usefu reaions beween he Aposo-Bernoui poynomias, Aposo-Euer poynomias and he Aposo-Genocchi poynomias. In his paper, we firs sudy severa eemenary properies of he generaized Lagrange-based Aposo ype Hermie poynomias T α1,,αr;α,k x 1 x r ; x, y. Some impici summaion formuae and genera symmery ideniies are derived by using differen anayica means and appying generaing funcions. 2. Definiions and Basic Properies of he Generaized Lagrange-Based Aposo-Type Poynomias T α1,,αr;α,k x 1 x r ; x, y In his secion, we inroduce Lagrange-based Aposo ype Hermie poynomias and give he expici represenaions and is basic properies of he generaized Lagrange-based Aposo-ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y.

4 Preprins NOT PEER-REVIEWED Posed: 19 Juy Definiion 2.1. The Lagrange-based Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y are defined by means of he foowing generaing funcion: 1 x α 1 2 k λe + 1 k+1 α e x+y2 λ C; α C 1,, r. T α 1,,α r ;α,k x 1 x r ; x, y n, 2.1 In he paricuar cases, when k 0 and k 1, we define he Lagrange-based Aposo Hermie-Bernoui poynomias H B α 1,,α r ;α x 1 x r ; x, y and he Lagrangebased Aposo Hermie-Genocchi poynomias H G α 1,,α r ;α x 1 x r ; x, y as foows. Definiion 2.2. The Lagrange-based Aposo Hermie-Bernoui poynomias H B α 1,,α r ;α x 1 x r ; x, y are defined by means of he foowing generaing funcion: α 1 x α λe e x+y2 1 1 λ C; α C 1,, r. HB α 1,,α r ;α x 1 x r ; x, y n, 2.2 Definiion 2.3. The Lagrange-based Aposo Hermie-Genocchi poynomias H G α 1,,α r ;α x 1 x r ; x, y are defined by means of he foowing generaing funcion: 1 x α 1 2 λe + 1 α e x+y2 HG α 1,,α r ;α x 1 x r ; x, y n, 2.3 λ C; α C 1,, r. Furhermore, we define he Lagrange-based Aposo Hermie-Euer poynomias as foows. Definiion 2.4. The Lagrange-based Aposo Hermie-Euer poynomias H E α1,,αr;α x 1 x r ; x, y are defined by means of he foowing generaing funcion: 1 x α 1 2 λe + 1 α e x+y2 HE α1,,αr;α x 1 x r ; x, y n, 2.4 λ C; α C 1,, r. Theorem 2.1. Each of he foowing reaionships hods rue: HB α1,,αr;α HE α 1,,α r ;α HG α1,,αr;α HB α 1,,α r ;α x 1,, x r ; x, 0 HE α 1,,α r ;α x 1,, x r ; x, 0 0,, 0; x, y H B n α x, y; λ ,, 0; x, y H E n α x, y; λ ,, 0; x, y H G α n x, y; λ. 2.7 B α 1,,α r ;α n x; λg α 1,,α r ;α x 1,, x r. 2.8 n! 0 E α 1,,α r ;α n x; λg α 1,,α r ;α x 1,, x r. 2.9 n! 0 HG α 1,,α r ;α x 1,, x r ; x, 0 G α1,,αr;α n x; λg α1,,αr;α x 1,, x r. n!

5 Preprins NOT PEER-REVIEWED Posed: 19 Juy 2018 HB α 1,,α r ;1 0,, 0; 0, 0 B n λ HE α 1,,α r ;1 0,, 0; 0, 0 E n λ HG α1,,αr;1 0,, 0; 0, 0 G n λ HB α 1,,α r ;0 x 1,, x r ; 0, 0 H E α 1,,α r ;0 x 1,, x r ; 0, and H G α 1,,α r ;0 x 1,, x r ; 0, 0 g α 1,,α r n x 1,, x r Theorem 2.2. The Lagrange-based Aposo Hermie Bernoui poynomias, he Lagrange-based Aposo Hermie Euer poynomias and he Lagrange-based Aposo Hermie Genocchi poynomias are expiciy given by and HB α 1,,α r ;α x 1,, x r ; x, y HE α 1,,α r ;α x 1,, x r ; x, y HG α1,,αr;α x 1,, x r ; x, y HB α n x, y; λgα 1,,α r x 1,, x r n! 0 HE α n x, y; λgα 1,,α r x 1,, x r n! 0 HG α n x, y; λgα 1,,α r x 1,, x r n! respecivey. Proof. I is fairy sraighforward o observe from 1.1 and 2.2 [ α ] HB α 1,,α r ;α x 1,, x r ; x, y λe e x+y2 1 n x α 1 HB n α x, y; λ n g α1,,αr 0 HB α n x, y; λgα1,,αr x 1,, x r n! Comparing he coefficien of n in boh sided, we ge he resu The expici represenaions of H E α 1,,α r ;α x 1,, x r ; x, y and H G α 1,,α r ;α x 1,, x r ; x, y foow in a simiar manner from he generaing funcion 1.1 in conuncion wih 2.3 and 2.4 respecivey. Theorem 2.3. The foowing ideniies hod rue for he Lagrange-based Aposo ype Hermie-Bernoui poynomias, he Lagrange-based Aposo ype Hermie-Euer poynomias and he Lagrange-based Aposo ype Hermie-Genocchi poynomias 0 0 T α1+β1,,αr+βr;α+β,k x 1 x r ; x + z, y + u T α 1,,α r ;α,λ,k x 1 x r ; x, yt β 1,,β r ;β n,λ,k x 1 x r ; z, u HB α 1+β 1,,α r +β r ;α+β x 1 x r ; x + z, y + u HB α 1,,α r ;α,λ x 1 x r ; x, y H B β 1,,β r ;β n,λ x 1 x r ; z, u HE α 1+β 1,,α r +β r ;α+β x 1 x r ; x + z, y + u n. 5

6 Preprins NOT PEER-REVIEWED Posed: 19 Juy and 0 0 HE α1,,αr;α,λ x 1 x r ; x, y H E β1,,βr;β n,λ x 1 x r ; z, u HG α 1+β 1,,α r +β r ;α+β x 1 x r ; x + z, y + u HG α 1,,α r ;α,λ x 1 x r ; x, y H G β 1,,β r ;β n,λ x 1 x r ; z, u Summaion formuae for generaized Lagrange-based Aposo-ype Hermie poynomias Firs, we prove he foowing resu invoving generaized Lagrange-based Aposoype Hermie poynomias by using series rearrangemen echniques. We now begin wih he foowing heorem. Theorem 3.1. The foowing impici summaion formuae for Lagrange-based Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y hods rue: T α 1,,α r ;α,k x 1 x r ; x, y T α 1,,α r ;α x 1 x r ; x z m 1 H n m z, y n m!. m0 Proof. By he expoiing generaing funcion 2.1 and using he x α 2 k α λe + 1 k+1 e x z e z+y2 1 m0 T α 1,,α r ;α x 1 x r ; x z m H n z, y n. 3.1 Now repacing n by n m and comparing he coefficiens of n, we ge he resu 3.1. Theorem 3.2. The foowing impici summaion formuae for Lagrange-based Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y hods rue: T α 1,,α r ;α,k x 1 x r ; x+u, y+w T α 1,,α r ;α x 1 x r ; x z m 1 H n m u, w n m!. m0 3.2 Proof. Appying he definiion 2.1, we have 1 x α 2 k α λe + 1 k+1 e x+u+y+w2 T α 1,,α r ;α,k x 1 x r ; x+u, y+w n 1 m0 T α 1,,α r ;α x 1 x r ; x, y m H n u, w n. Repacing n by n m in he r.h.s and comparing he coefficien of n, we ge he resu 3.2. Theorem 3.3. The foowing impici summaion formuae for Lagrange-based Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y hods rue:

7 Preprins NOT PEER-REVIEWED Posed: 19 Juy T α 1,,α r ;α,k x 1 x r ; x, y n 2 [ n 2 ] m0 0 Proof. Using he definiion 2.1, we have T α 1,,α r ;α,k x 1 x r ; x, y n m0 m0 T α 1,,α r ;α x n m 2 y 1 n m 2! x α 2 k α λe + 1 k+1 e x+y2 1 T α 1,,α r ;α n x n n 0 T α1,,αr;α x n m n n m! Repacing n by n 2 in above equaion, we ge n 2 [ n 2 ] m0 0 y 2! 0 y 2!. T α1,,αr;α x n m 2 y n n m 2!. Comparing he coefficien of n, we ge he resu 3.3. Theorem 3.4. The foowing impici summaion formuae for Lagrange-based Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y hods rue: T α 1,,α r ;α,k x 1 x r ; x + 1, y T α 1,,α r ;α x 1 x r ; x, y m 1 n m!. 3.4 m0 Proof. By he definiion of generaized Lagrange-based Aposo ype poynomias, we have T α1,,αr;α,k x 1 x r ; x + 1, y n T α1,,αr;α,k x 1 x r ; x, y n m0 1 T α 1,,α r ;α 2 k α 1 x α λe + 1 k+1 e x+y2 e 1 x 1 x r ; x, y m n T α 1,,α r ;α,k x 1 x r ; x, y n. Repacing n by n m, we have T α 1,,α r ;α x 1 x r ; x, y m n n m! m0 Equaing heir coefficiens of n eads o formua Genera symmery ideniies T α 1,,α r ;α,k x 1 x r ; x, y n. In his secion, we esabish genera symmery ideniies for he generaized Lagrangebased Aposo ype Hermie poynomias T α 1,,α r ;α,k x 1 x r ; x, y and generaized Lagrange-based Aposo ype poynomias T α1,,αr;α,k x 1 x r ; x by appying he generaing funcions 2.1 and 1.6. Such ype of works, inroduced here by he approach given in he recen works of Khan [5] and Pahan and Khan [13, 14, 15].

8 Preprins NOT PEER-REVIEWED Posed: 19 Juy Theorem 4.1. Le a, b > 0 and a b. Then for x, y R and n 0, he foowing ideniy hods rue: a n m b m T α1,,αr;α n x 1 x r ; bx, b 2 yt α1,,αr;α x 1 x r ; ax, a 2 y m0 m0 b n m a m T α 1,,α r ;α n x 1 x r ; ax, a 2 yt α 1,,α r ;α x 1 x r ; bx, b 2 y. 4.1 Proof. Sar wih G 1 x ab α 2 2k α ab λe a + 1 k+1 λe b + 1 k+1 e abx+a2 b 2 y Then he expression for G is symmeric in a and b and we can expand G ino series in wo ways o obain: G T α1,,αr;α,k x 1 x r ; bx, b 2 ya n T α1,,αr;α x 1 x r ; ax, a 2 yb m m0 m0 a n m b m T α 1,,α r ;α n x 1 x r ; bx, b 2 yt α 1,,α r ;α x 1 x r ; ax, a 2 y n. 4.3 On he simiar ines we can show ha G x 1 x r ; ax, a 2 ya n m0 T α1,,αr;α,k m0 T α1,,αr;α x 1 x r ; bx, b 2 yb m b n m a m T α 1,,α r ;α n x 1 x r ; ax, a 2 yt α 1,,α r ;α x 1 x r ; bx, b 2 y n. 4.4 By comparing he coefficiens of n on he righ hand sides of he as wo equaions, we arrive he desired resu. Remark 4.1. For α 1, he above resu reduces o a n m b m T α 1,,α r n x 1 x r ; bx, b 2 yt α 1,,α r x 1 x r ; ax, a 2 y m0 m0 b n m a m T α1,,αr n x 1 x r ; ax, a 2 yt α1,,αr x 1 x r ; bx, b 2 y. 4.5 Remark 4.2. By seing b 1 in Theorem 4.1, we immediaey ge he foowing resu a n m T α 1,,α r ;α n x 1 x r ; x, yt α 1,,α r ;α x 1 x r ; ax, a 2 y m0 m0 a m T α 1,,α r ;α n x 1 x r ; ax, a 2 yt α 1,,α r ;α x 1 x r ; x, y. 4.6 Theorem 4.2. Le a, b, c > 0 and a b. Then for x, y R and n 0, he foowing ideniy hods rue: a 1 b 1 λ i+ a n m b m α1,,αr,α T bx + b a i +, b2 α1,,αr,α z T ay m0 i0 0,k

9 Preprins NOT PEER-REVIEWED Posed: 19 Juy 2018 b 1 a 1 λ i+ a m b n m T m0 i0 0 α1,,αr,α,k ax + a b i +, a2 z T α1,,αr,α 4.7 Proof. Le 2 2k ab 2 α 1 + λ 1 a+1 e ab H 1 x ab α λe a + 1 k+1 λe b + 1 k+1 λe a + 1λe b + 1 eabx+y+a 1 g by. 1 x a α 2 k α a λe a + 1 k+1 e abx+a2 b 2 z 2 1 λe b a λe b x b α 2 k α b 1 λe λe b + 1 k+1 e aby a b λe a k α a 1 a 1 x a α λe a + 1 k+1 e abx+a2 b 2 z 2 λ i e bi i0 1 x b α 2 k α b b 1 λe b + 1 k+1 e aby λ e a 1 1 x a α 2 k α a 1 a b 1 λe a + 1 k+1 e a2 b 2 z 2 λ i+ e bx+ b a i+a 1 m0 a 1 b 1 λ i+ T α 1,,α r,α i0 0 m0 i0 0,k T α 1,,α r,α ayb m. bx + b a i +, b2 z a 1 b 1 λ i+ a n m b m T α 1,,α r,α,k 0 i0 0 a n m0 bx + b a i +, b2 z 9 T α 1,,α r,α ayb m T α 1,,α r,α ay n. 4.8 Since 1 a+1 1 b+1, he expression for 2 2k ab 2 α 1 + λ 1 a+1 e ab H 1 x ab α λe a + 1 k+1 λe b + 1 k+1 λe a + 1λe b + 1 eabx+y+a 1 is symmeric in a and b. Therefore, by symmery we obain he foowing power series expansion for H b 1 a 1 λ i+ a m b n m T α 1,,α r,α m0 i0 0,k ax + a b i +, a2 z T α 1,,α r,α by n. 4.9 By comparing he coefficiens of n on he righ hand sides of he as wo equaions,we arrive a he desired resu. 2 b 2 z 2. 2 b 2 z 2. References [1] Andrews, L. C, Specia funcions for engineers and mahemaicians, Macmian Co. New York, [2] Be, E. T, Exponenia poynomias, Ann. of Mah., ,

10 Preprins NOT PEER-REVIEWED Posed: 19 Juy [3] Chan, C. W.-Ch, Chyan, Ch. J and Srivasava, H. M, The Lagrange poynomias in severa variabes, Ineg. Trans. Spec. Func., , [4] Daoi, G, Lorenzua, S and Cesarano, C, Finie sums and generaized forms of Bernoui poynomias, Rendiconi di Mahemaica, , [5] Khan, W. A: Some properies of he generaized Aposo ype Hermie-based poynomias, Kyungpook Mah. J., , [6] Lu, D. Q and Luo, Q. M, Some properies of he generaized Aposo pe poynomias, Boundary Vaue Probems., 2013, 2013:64. [7] Luo, Q. M, Aposo Euer poynomias of higher order and gaussian hypergeomeric funcions, Taiwanese J. Mah., , [8] Luo, Q. M, q-exensions for he Aposo-Genocchi poynomias, Gen. Mah., , [9] Luo, Q. M, Exensions for he Genocchi poynomias and is fourier expansions and inegra represenaions, Osaka J. Mah., , [10] Luo, Q. M and Srivasava, H. M, Some generaizaions of he Aposo-Bernoui and Aposo-Euer poynomias, J. Mah. Ana. App., , [11] Luo, Q. M and Srivasava, H. M, Some generaizaions of he Aposo Genocchi poynomias and he Siring number of he second kind, App. Mah. Compu., , [12] Luo, Q. M and Srivasava, H. M, Some reaionships beween he Aposo- Bernoui and Aposo-Euer poynomias, Compu. Mah. App., , [13] Pahan, M. A and Khan, W. A, Some impici summaion formuas and symmeric ideniies for he generaized Hermie-based poynomias, Aca Universiais Apuensis, , [14] Pahan, M. A and Khan, W. A, Some impici summaion formuas and symmeric ideniies for he generaized Hermie-Bernoui poynomias, Medierr. J. Mah., , [15] Pahan, M. A and Khan, W. A, A new cass of generaized poynomias associaed wih Hermie and Euer poynomias, Medierr. J. Mah., , [16] Srivasava, H. M, Ozarsan, M. A, Kaanogu, C, Some generaized Lagrangebased Aposo Bernoui, Aposo Euer and Aposo Genocchi poynomias, Russian J. Mah. Phy., ,