Introduction and motivation Outline Summation formula for generalized discrete q -Hermite II polynomials AIMS-Volkswagen Workshop, Douala October 5-12, 2018 African Institute for Mathematical Sciences, Cameroon By Sama Arjika Faculty of Sciences and Technics University of Agadez, Niger 12 septembre 2018 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation The classical orthogonal polynomial (COP) and the quantum orthogonal polynomials (QOP) (also called q -orthogonal polynomials) constitute an interesting set of special functions. They appear in several branches of sciences such as : continued fractions, Eulerian 1 series, theta functions, elliptic functions, · · · [Andrews (1986), Fine (1988)], quantum groups and quantum algebras [Gasper and Rahman (1990), 2 Koornwinder (1990) and (1994), Nikiforov et al (1991), Vilenkin and Klimyk (1992)]. They have been intensively studied in the last years by several people, [Koekoek and Swarttouw (1998), Lesky (2005), Koekoek et al (2010)], · · · . Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation Each family of COP and QOP occupy different levels within the so-called, Askey-Wilson scheme and are characterized by the properties : they are solutions of a hypergeometric second order differential 1 equation, they are generated by a recursion relation, 2 they are orthogonal with respect to a weight function, 3 they obey the Rodrigues-type formula. 4 In this scheme, the Hermite polynomials are the ground level and most of there properties can be generalized. Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation In their paper, ` Alvarez-Nodarse et al [Int. J. Pure. Appl. Math. 10 (3) 331-342 (2014)], have introduced a q -extension of the discrete q -Hermite II polynomials as : H ( µ ) ( − 1) n ( q ; q ) n L ( µ − 1 / 2) ( x 2 ; q ) 2 n ( x ; q ) : = n (1) H ( µ ) ( − 1) n ( q ; q ) n x L ( µ +1 / 2) ( x 2 ; q ) 2 n +1 ( x ; q ) : = n where µ > − 1 / 2 , L ( α ) n ( x ; q ) are the q -Laguerre polynomials given by � q − n n ( x ; q ) := ( q α +1 ; q ) n � � L ( α ) � q ; − q n + α +1 x 1 φ 1 . (2) � q α +1 ( q ; q ) n For µ = 0 in (1), the polynomials H (0) n ( x ; q ) correspond to the discrete q -Hermite II polynomial n ( x ; q 2 ) = q n ( n − 1) ˜ H (0) h n ( x ; q ) . Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation Alvarez-Nodarse et al showed that the polynomials H ( µ ) ` n ( x ; q ) satisfy the orthogonality relation � ∞ H ( µ ) n ( x ; q ) H ( µ ) m ( x ; q ) ω ( x ) dx = π q − n / 2 ( q 1 / 2 ; q 1 / 2 ) n ( q 1 / 2 ; q ) 1 / 2 δ nm −∞ on the whole real line R with respect to the positive weight function ω ( x ) = 1 / ( − x 2 ; q ) ∞ . A detailed discussion of the properties of the polynomials H ( µ ) n ( x ; q ) can be found in [Int. J. Pure. Appl. Math. 10 (3) 331-342 (2014)]. Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation Recently, Saley Jazmat et al [Bulletin of Mathematical Ana. App. 6 (4), 16-43 (2014)], introduced a novel extension of discrete q -Hermite II polynomials by using new q -operators. This extension is defined as : ( q ; q ) 2 n ( − 1) n q − n (2 n − 1) ˜ L ( α ) x 2 q − 2 α − 1 ; q 2 � � h 2 n ,α ( x ; q ) = n ( q 2 α +2 ; q 2 ) n (3) ( q ; q ) 2 n +1 ( − 1) n q − n (2 n +1) ˜ x L ( α +1) x 2 q − 2 α − 1 ; q 2 � � h 2 n +1 ,α ( x ; q ) = . n ( q 2 α +2 ; q 2 ) n +1 For α = − 1 / 2 in (3), the polynomials ˜ h n , − 1 2 ( x ; q ) correspond to the discrete q -Hermite II polynomials, i.e., ˜ 2 ( x ; q ) = ˜ h n , − 1 h n ( x ; q ) . Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation The generalized discrete q -Hermite II polynomials ˜ h n ,α ( x ; q ) satisfy the orthogonality relation � + ∞ ˜ h n ,α ( x ; q )˜ h m ,α ( x ; q ) ω α ( x ; q ) | x | 2 α +1 d q x −∞ = 2 q − n 2 (1 − q )( − q , − q , q 2 ; q 2 ) ∞ ( q ; q ) 2 n δ n , m (4) ( − q − 2 α − 1 , − q 2 α +3 , q 2 α +2 ; q 2 ) ∞ ( q ; q ) n ,α on the real line R with respect to the positive weight function ω α ( x ) = 1 / ( − q − 2 α − 1 x 2 ; q 2 ) ∞ . Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Introduction and motivation Motivated by Saley Jazmat’s work [Bul. Math. Anal. App. 6 (4), 16-43 (2014)], our interest in this work is to introduce new family of “ generalized discrete q-Hermite II 1 polynomials (in short gdq-H2P) ˜ h n ,α ( x , y | q )” which is an extension of the generalized discrete q -Hermite II polynomials ˜ h n ,α ( x ; q ), and investigate summation formula. 2 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Outline Notations and definitions 1 Generalized discrete q -Hermite II polynomials { ˜ h n ,α ( x , y | q ) } ∞ 2 n =0 Connection formulae for the generalized discrete q -Hermite II 3 polynomials { ˜ h n ,α ( x , y | q ) } ∞ n =0 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Outline Notations and definitions 1 Generalized discrete q -Hermite II polynomials { ˜ h n ,α ( x , y | q ) } ∞ 2 n =0 Connection formulae for the generalized discrete q -Hermite II 3 polynomials { ˜ h n ,α ( x , y | q ) } ∞ n =0 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Introduction and motivation Outline Outline Notations and definitions 1 Generalized discrete q -Hermite II polynomials { ˜ h n ,α ( x , y | q ) } ∞ 2 n =0 Connection formulae for the generalized discrete q -Hermite II 3 polynomials { ˜ h n ,α ( x , y | q ) } ∞ n =0 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Notations and definitions hn ,α ( x , y | q ) }∞ Generalized discrete q -Hermite II polynomials { ˜ n =0 hn ,α ( x , y | q ) }∞ Connection formulae for the generalized discrete q -Hermite II polynomials { ˜ n =0 Conclusion Notations and definitions Throughout this paper, we assume that 0 < q < 1 , α > − 1. For a complex number a , ⋆ the q -shifted factorials are defined by : n − 1 ∞ � � (1 − aq k ) , n ≥ 1; ( a ; q ) ∞ = (1 − aq k ) . ( a ; q ) 0 = 1; ( a ; q ) n = k =0 k =0 ⋆ The q -number is defined by : n [ n ] q = 1 − q n � 1 − q , n ! q := [ k ] q , 0! q := 1 , n ∈ N . (5) k =1 Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Notations and definitions hn ,α ( x , y | q ) }∞ Generalized discrete q -Hermite II polynomials { ˜ n =0 hn ,α ( x , y | q ) }∞ Connection formulae for the generalized discrete q -Hermite II polynomials { ˜ n =0 Conclusion Notations and definitions Hahn q -addition and q -subtraction For x , y ∈ R , ⋆ the Hahn q -addition ⊕ q is defined by : � n : ( x + y )( x + qy ) . . . ( x + q n − 1 y ) � x ⊕ q y = n q ( k 2 ) x n − k y k � = ( q ; q ) n , n ≥ 1 , (6) ( q ; q ) k ( q ; q ) n − k k =0 � 0 := 1. � and x ⊕ q y ⋆ The q -subtraction ⊖ q is given by � n := � n � � x ⊖ q y x ⊕ q ( − y ) (7) � 0 := 1. � and x ⊖ q y Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
Notations and definitions hn ,α ( x , y | q ) }∞ Generalized discrete q -Hermite II polynomials { ˜ n =0 hn ,α ( x , y | q ) }∞ Connection formulae for the generalized discrete q -Hermite II polynomials { ˜ n =0 Conclusion Notations and definitions The generalized backward and forward q -derivative operators D q ,α 1 and D + q ,α , Saley Jazmat et al are defined : D q ,α f ( x ) = f ( x ) − q 2 α +1 f ( qx ) q ,α f ( x ) = f ( q − 1 x ) − q 2 α +1 f ( x ) , D + . (1 − q ) x (1 − q ) x Remark that, for α = − 1 2 , we have D q ,α = D q , D + q ,α = D + q where 2 D q and D + q are the Jackson’s q -derivative with q f ( x ) = f ( q − 1 x ) − f ( x ) D q f ( x ) = f ( x ) − f ( qx ) , D + . (8) (1 − q ) x (1 − q ) x For f ( x ) = x n , we have 3 D q ,α x n = [ n ] q ,α x n − 1 , q ,α x n = q − n [ n ] q ,α x n − 1 D + where [ n ] q ,α := [ n + 2 α + 1] q , [ n ] q , − 1 / 2 = [ n ] q . Sama Arjika Summation formula for generalized discrete q -Hermite II polynomials
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