[PPT] - Summation formula for generalized discrete q -Hermite II polynomials PowerPoint Presentation

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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

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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

1

several branches of sciences such as : continued fractions, Eulerian series, theta functions, elliptic functions,· · · [Andrews (1986), Fine (1988)],

2

quantum groups and quantum algebras [Gasper and Rahman (1990), 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

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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 :

1

they are solutions of a hypergeometric second order differential equation,

2

they are generated by a recursion relation,

3

they are orthogonal with respect to a weight function,

4

they obey the Rodrigues-type formula. 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

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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)

2n (x; q) :

= (−1)n(q; q)n L(µ−1/2)

n

(x2; q) (1) H(µ)

2n+1(x; q) :

= (−1)n(q; q)n x L(µ+1/2)

n

(x2; q) where µ > −1/2, L(α)

n (x; q) are the q-Laguerre polynomials given by

L(α)

n (x; q) := (qα+1; q)n

(q; q)n

1φ1

q−n qα+1

q; −qn+α+1x
.

(2) For µ = 0 in (1), the polynomials H(0)

n (x; q) correspond to the discrete

q-Hermite II polynomial H(0)

n (x; q2) = qn(n−1)˜

hn(x; q).

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Introduction and motivation Outline

Introduction and motivation ` Alvarez-Nodarse et al showed that the polynomials H(µ)

n (x; q) satisfy the

rthogonality relation

∞

−∞

H(µ)

n (x; q)H(µ) m (x; q)ω(x)dx = π q−n/2(q1/2; q1/2)n(q1/2; q)1/2 δnm

n the whole real line R with respect to the positive weight function

ω(x) = 1/(−x2; 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

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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 : ˜ h2n,α(x; q) = (−1)n q−n(2n−1) (q; q)2n (q2α+2; q2)n L(α)

n

x2q−2α−1; q2

(3) ˜ h2n+1,α(x; q) = (−1)n q−n(2n+1) (q; q)2n+1 (q2α+2; q2)n+1 x L(α+1)

n

x2q−2α−1; q2

. For α = −1/2 in (3), the polynomials ˜ hn,− 1

2 (x; q) correspond to the

discrete q-Hermite II polynomials, i.e., ˜ hn,− 1

2 (x; q) = ˜

hn(x; q).

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Introduction and motivation Outline

Introduction and motivation The generalized discrete q-Hermite II polynomials ˜ hn,α(x; q) satisfy the

rthogonality relation

+∞

−∞

˜ hn,α(x; q)˜ hm,α(x; q)ωα(x; q)|x|2α+1dqx = 2q−n2 (1 − q)(−q, −q, q2; q2)∞ (−q−2α−1, −q2α+3, q2α+2; q2)∞ (q; q)2

n

(q; q)n,α δn,m (4)

n the real line R with respect to the positive weight function

ωα(x) = 1/(−q−2α−1 x2; q2)∞.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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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

1

to introduce new family of “generalized discrete q-Hermite II polynomials (in short gdq-H2P) ˜ hn,α(x, y|q)” which is an extension

f the generalized discrete q-Hermite II polynomials ˜

hn,α(x; q),

2

and investigate summation formula.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Introduction and motivation Outline

Outline

1

Notations and definitions

2

Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 3

Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Introduction and motivation Outline

Outline

1

Notations and definitions

2

Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 3

Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Introduction and motivation Outline

Outline

1

Notations and definitions

2

Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 3

Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ 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 : (a; q)0 = 1; (a; q)n =

n−1

k=0

(1 − aqk), n ≥ 1; (a; q)∞ =

∞

k=0

(1 − aqk). ⋆ The q-number is defined by : [n]q = 1 − qn 1 − q , n!q :=

n

k=1

[k]q, 0!q := 1, n ∈ N. (5)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Notations and definitions

Hahn q-addition and q-subtraction For x, y ∈ R, ⋆ the Hahn q-addition ⊕q is defined by :

x ⊕q y

n : = (x + y)(x + qy) . . . (x + qn−1y) = (q; q)n

n

k=0

q(k

2)xn−ky k

(q; q)k(q; q)n−k , n ≥ 1, (6) and

x ⊕q y

0 := 1. ⋆ The q-subtraction ⊖q is given by

x ⊖q y

n :=

x ⊕q (−y)

n (7) and

x ⊖q y

0 := 1.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Notations and definitions

1

The generalized backward and forward q-derivative operators Dq,α and D+

q,α, Saley Jazmat et al are defined :

Dq,αf (x) = f (x) − q2α+1f (qx) (1 − q)x , D+

q,αf (x) = f (q−1x) − q2α+1f (x)

(1 − q)x .

2

Remark that, for α = − 1

2, we have Dq,α = Dq, D+ q,α = D+ q where

Dq and D+

q are the Jackson’s q-derivative with

Dqf (x) = f (x) − f (qx) (1 − q)x , D+

q f (x) = f (q−1x) − f (x)

(1 − q)x . (8)

3

For f (x) = xn, we have Dq,αxn = [n]q,αxn−1, D+

q,αxn = q−n[n]q,αxn−1

where [n]q,α := [n + 2α + 1]q, [n]q,−1/2 = [n]q.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized q-shifted factorials The generalized q-shifted factorials are defined as : (n + 1)!q,α = [n + 1 + θn(2α + 1)]q n!q,α (9) (q; q)n+1,α = (1 − q)[n + 1 + θn(2α + 1)]q(q; q)n,α, (10) where θn = 1 if n even if n odd. ⋆ Remark that, for α = −1/2, we have (q; q)n,−1/2 = (q; q)n, n!q,−1/2 = (q; q)n (1 − q)n . (11) ⋆ We denote (q; q)2n,α = (q2; q2)n(q2 α+2; q2)n, (12) (q; q)2n+1,α = (q2; q2)n(q2 α+2; q2)n+1. (13)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized q-exponential functions The two Euler’s q-analogs of the exponential functions are given by eq(x) :=

∞

n=0

xn (q; q)n = 1 (x; q)∞ (14) and Eq (x) :=

∞

n=0

q(

n 2)

(q; q)n xn = (−x; q)∞. (15) For m ≥ 1, we define two generalized q-exponential functions as follows ˜ Eqm,α(x) :=

∞

k=0

qmk(k−1)/2 xk (qm; qm)k,α , (16) and ˜ eqm,α(x) :=

∞

k=0

xk (qm; qm)k,α , |x| < 1. (17)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized q-exponential functions The two Euler’s q-analogs of the exponential functions are given by eq(x) :=

∞

n=0

xn (q; q)n = 1 (x; q)∞ (14) and Eq (x) :=

∞

n=0

q(

n 2)

(q; q)n xn = (−x; q)∞. (15) For m ≥ 1, we define two generalized q-exponential functions as follows ˜ Eqm,α(x) :=

∞

k=0

qmk(k−1)/2 xk (qm; qm)k,α , (16) and ˜ eqm,α(x) :=

∞

k=0

xk (qm; qm)k,α , |x| < 1. (17)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Particular case Remark that, for m = 1 and α = − 1

2, we have :

˜ Eq,− 1

2 (x) = Eq(x),

˜ eq,− 1

2 (x) = eq(x).

(18) Elementary result For m = 2, the following elementary result is useful in the sequel to establish summation formula for gdq-H2P : ˜ eq2,− 1

2 (x)˜

Eq2,− 1

2 (y) = ˜

eq2,− 1

2 (x ⊕q2 y),

(19) ˜ eq,− 1

2 (x)˜

Eq2,− 1

2 (−y) = ˜

eq(x ⊖q,q2 y), ˜ eq2,− 1

2 (x)˜

Eq2,− 1

2 (−x) = 1,

(20) where (a ⊖q,q2 b)n := n!q

n

k=0

(−1)kqk(k−1) (n − k)!q k!q2 an−kbk, (a ⊖q,q2 b)0 := 1.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Particular case Remark that, for m = 1 and α = − 1

2, we have :

˜ Eq,− 1

2 (x) = Eq(x),

˜ eq,− 1

2 (x) = eq(x).

(18) Elementary result For m = 2, the following elementary result is useful in the sequel to establish summation formula for gdq-H2P : ˜ eq2,− 1

2 (x)˜

Eq2,− 1

2 (y) = ˜

eq2,− 1

2 (x ⊕q2 y),

(19) ˜ eq,− 1

2 (x)˜

Eq2,− 1

2 (−y) = ˜

eq(x ⊖q,q2 y), ˜ eq2,− 1

2 (x)˜

Eq2,− 1

2 (−x) = 1,

(20) where (a ⊖q,q2 b)n := n!q

n

k=0

(−1)kqk(k−1) (n − k)!q k!q2 an−kbk, (a ⊖q,q2 b)0 := 1.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Discrete q-Hermite II polynomials ˜ hn(x|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k (q; q)n−2k (q2; q2)k . (21) For α > −1, we define a sequence of generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 as follows :

Definition For x, y ∈ R, a gdq-H2P{˜ hn,α(x, y|q)}∞

n=0 are defined by :

˜ hn,α(x, y|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k y k (q; q)n−2k,α (q2; q2)k (22) and ˜ hn,α(x, 0|q) := [(q; q)n/(q; q)n,α]xn. (23)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Discrete q-Hermite II polynomials ˜ hn(x|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k (q; q)n−2k (q2; q2)k . (21) For α > −1, we define a sequence of generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 as follows :

Definition For x, y ∈ R, a gdq-H2P{˜ hn,α(x, y|q)}∞

n=0 are defined by :

˜ hn,α(x, y|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k y k (q; q)n−2k,α (q2; q2)k (22) and ˜ hn,α(x, 0|q) := [(q; q)n/(q; q)n,α]xn. (23)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Discrete q-Hermite II polynomials ˜ hn(x|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k (q; q)n−2k (q2; q2)k . (21) For α > −1, we define a sequence of generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 as follows :

Definition For x, y ∈ R, a gdq-H2P{˜ hn,α(x, y|q)}∞

n=0 are defined by :

˜ hn,α(x, y|q) := (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k y k (q; q)n−2k,α (q2; q2)k (22) and ˜ hn,α(x, 0|q) := [(q; q)n/(q; q)n,α]xn. (23)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Particular cases of gdq-H2H ˜ hn,α(x, y|q)

1

For y = 1, we have ˜ hn,α(x, 1|q) = ˜ hn,α(x; q) (24) where ˜ hn,α(x; q) is the generalized discrete q-Hermite II polynomial.

2

For α = −1/2 and y = 1, we have ˜ hn,−1/2(x, 1|q) = ˜ hn(x; q). (25) where ˜ hn(x; q) is the discrete q-Hermite II polynomial.

3

Indeed since lim

q→1

(qa; q)n (1 − q)n = (a)n, one readily verifies that lim

q→1

˜ hn,− 1

2 (

1 − q2x, 1|q)

(1 − q2)n/2 = h

α+ 1

2

n

(x) 2n (26) where h

α+ 1

2

n

(x) is the Rosenblums generalized Hermite polynomial.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Particular cases of gdq-H2H ˜ hn,α(x, y|q)

1

For y = 1, we have ˜ hn,α(x, 1|q) = ˜ hn,α(x; q) (24) where ˜ hn,α(x; q) is the generalized discrete q-Hermite II polynomial.

2

For α = −1/2 and y = 1, we have ˜ hn,−1/2(x, 1|q) = ˜ hn(x; q). (25) where ˜ hn(x; q) is the discrete q-Hermite II polynomial.

3

Indeed since lim

q→1

(qa; q)n (1 − q)n = (a)n, one readily verifies that lim

q→1

˜ hn,− 1

2 (

1 − q2x, 1|q)

(1 − q2)n/2 = h

α+ 1

2

n

(x) 2n (26) where h

α+ 1

2

n

(x) is the Rosenblums generalized Hermite polynomial.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

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Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Particular cases of gdq-H2H ˜ hn,α(x, y|q)

1

For y = 1, we have ˜ hn,α(x, 1|q) = ˜ hn,α(x; q) (24) where ˜ hn,α(x; q) is the generalized discrete q-Hermite II polynomial.

2

For α = −1/2 and y = 1, we have ˜ hn,−1/2(x, 1|q) = ˜ hn(x; q). (25) where ˜ hn(x; q) is the discrete q-Hermite II polynomial.

3

Indeed since lim

q→1

(qa; q)n (1 − q)n = (a)n, one readily verifies that lim

q→1

˜ hn,− 1

2 (

1 − q2x, 1|q)

(1 − q2)n/2 = h

α+ 1

2

n

(x) 2n (26) where h

α+ 1

2

n

(x) is the Rosenblums generalized Hermite polynomial.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 26

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Recursion relation The recursion relation for gdq-H2P {˜ hn,α(x, y|q)}∞

n=0 holds true.

x˜ hn,α(x, y|q) − y q−2n+1(1 − qn)˜ hn−1,α(x, y|q) = 1 − qn+1+θn(2α+1) 1 − qn+1 ˜ hn+1,α(x, y|q).

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 27

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Theorem 1 We have : lim

α→+∞

˜ h2n,α(x, y|q) = q−n(2n−1)(q; q)2n (−y)n Sn

x2y −1q−1; q2

(27) and lim

α→+∞

˜ h2n+1,α(x, y|q) = q−n(2n+1)(q; q)2n+1 x (−y)n Sn

x2y −1q−1; q2

(28) where Sn(x; q) are the Stieltjes-Wigert polynomials.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 28

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Lemma For α > −1, the sequence of gdq-H2P {˜ hn,α(x, y|q)}∞

n=0 can be written

in terms of q-Laguerre polynomials L(α)

n (x; q) as

˜ h2n,α(x, y|q) = q−n(2n−1) (q; q)2n (q2α+2; q2)n (−y)n L(α)

n

x2y −1q−2α−1; q2

(29) and ˜ h2n+1,α(x, y|q) = q−n(2n+1) (q; q)2n+1 (q2α+2; q2)n+1 x (−y)n L(α+1)

n

x2y −1q−2α−1; q2

. (30)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 29

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Generalized discrete q-Hermite II polynomials

Proposition For α > −1, the sequence of gdq-H2P {˜ hn,α(x, y|q)}∞

n=0 can be written

in terms of basic hypergeometric functions as ˜ hn,α(x, y|q) = (q; q)n (q; q)n,α xn

2φ1

q−n, q−n−2α

q2; −y q2α+3

x2

.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 30

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞

n=0 Theorem 2 The sequence of gdq-H2P {˜ hn,α(x, y|q)}∞

n=0, satisfies the connection

formula ˜ hn,α(x, ω|q) = (q; q)n

⌊ n/2 ⌋

k=0

q−2nk+k(2k+1) (−ω ⊕q2 y)k (q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q). (31)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 31

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

To prove the above Theorem 2, we need the following generating function ˜ eq2,− 1

2 (−yt2)˜

Eq,α(xt) =

∞

n=0

q(n

2) tn

(q; q)n ˜ hn,α(x, y|q), |yt| < 1. (32) Replacing t by u ⊕q t in the last generating function, we have ˜ Eq,α

(u ⊕q t)x
˜

eq2,− 1

2

− y(u ⊕q t)2

=

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (33) which can be written as ˜ Eq,α

(u ⊕q t)x
= ˜

Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (34)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 32

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

To prove the above Theorem 2, we need the following generating function ˜ eq2,− 1

2 (−yt2)˜

Eq,α(xt) =

∞

n=0

q(n

2) tn

(q; q)n ˜ hn,α(x, y|q), |yt| < 1. (32) Replacing t by u ⊕q t in the last generating function, we have ˜ Eq,α

(u ⊕q t)x
˜

eq2,− 1

2

− y(u ⊕q t)2

=

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (33) which can be written as ˜ Eq,α

(u ⊕q t)x
= ˜

Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (34)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 33

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

To prove the above Theorem 2, we need the following generating function ˜ eq2,− 1

2 (−yt2)˜

Eq,α(xt) =

∞

n=0

q(n

2) tn

(q; q)n ˜ hn,α(x, y|q), |yt| < 1. (32) Replacing t by u ⊕q t in the last generating function, we have ˜ Eq,α

(u ⊕q t)x
˜

eq2,− 1

2

− y(u ⊕q t)2

=

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (33) which can be written as ˜ Eq,α

(u ⊕q t)x
= ˜

Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (34)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 34

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

To prove the above Theorem 2, we need the following generating function ˜ eq2,− 1

2 (−yt2)˜

Eq,α(xt) =

∞

n=0

q(n

2) tn

(q; q)n ˜ hn,α(x, y|q), |yt| < 1. (32) Replacing t by u ⊕q t in the last generating function, we have ˜ Eq,α

(u ⊕q t)x
˜

eq2,− 1

2

− y(u ⊕q t)2

=

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (33) which can be written as ˜ Eq,α

(u ⊕q t)x
= ˜

Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (34)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 35

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

Replacing y by ω and using various identities, we get :

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, ω|q) = ˜ eq2,− 1

2

− ω(u ⊕q t)2

˜ Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). The r.h.s of the last expression can be written as ˜ eq2,− 1

2

(−ω ⊕q2 y)(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (35)

r

∞

r=0

(−ω ⊕q2 y)r(u ⊕q t)2r (q2; q2)r

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (36)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 36

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

Replacing y by ω and using various identities, we get :

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, ω|q) = ˜ eq2,− 1

2

− ω(u ⊕q t)2

˜ Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). The r.h.s of the last expression can be written as ˜ eq2,− 1

2

(−ω ⊕q2 y)(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (35)

r

∞

r=0

(−ω ⊕q2 y)r(u ⊕q t)2r (q2; q2)r

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (36)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 37

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

Replacing y by ω and using various identities, we get :

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, ω|q) = ˜ eq2,− 1

2

− ω(u ⊕q t)2

˜ Eq2,− 1

2

y(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). The r.h.s of the last expression can be written as ˜ eq2,− 1

2

(−ω ⊕q2 y)(u ⊕q t)2 ∞
n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q) (35)

r

∞

r=0

(−ω ⊕q2 y)r(u ⊕q t)2r (q2; q2)r

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, y|q). (36)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 38

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

Let us substitute n + 2r = k = ⇒ r ≤ ⌊ k/2 ⌋ in the last equation, we get :

∞

n=0

 

⌊ n/2 ⌋

k=0

(q(

n−2k 2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q)   (u ⊕q t)n. (37) Summarizing the above calculations, we obtain

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, ω|q) =

∞

n=0

 

⌊ n/2 ⌋

k=0

(q(

n−2k 2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q)   (u ⊕q t)n. (38) By equating the coefficients of like powers of (u ⊕q t)n/(q; q)n in the last equation, we get the desired identity.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 39

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Proof. Summation formula

Let us substitute n + 2r = k = ⇒ r ≤ ⌊ k/2 ⌋ in the last equation, we get :

∞

n=0

 

⌊ n/2 ⌋

k=0

(q(

n−2k 2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q)   (u ⊕q t)n. (37) Summarizing the above calculations, we obtain

∞

n=0

q(n

2)(u ⊕q t)n

(q; q)n ˜ hn,α(x, ω|q) =

∞

n=0

 

⌊ n/2 ⌋

k=0

(q(

n−2k 2 )(−ω ⊕q2 y)k

(q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q)   (u ⊕q t)n. (38) By equating the coefficients of like powers of (u ⊕q t)n/(q; q)n in the last equation, we get the desired identity.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 40

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Connection formulae for the gdq-H2P {˜ hn,α(x, y|q)}∞

n=0 Particular cases Letting : (i) y = 0 in the assertion of Theorem 2, we get the definition of gdq-H2P, i.e., ˜ hn,α(x, ω|q) = (q; q)n

⌊ n/2 ⌋

k=0

(−1)kq−2nk+k(2k+1) xn−2k ωk (q2; q2)k (q; q)n−2k,α ; (39) (ii) ω = 0 in the assertion of Theorem 2, we get the inversion formula for gdq-H2P xn = (q; q)n,α

⌊ n/2 ⌋

k=0

q−2nk+3k2 y k (q2; q2)k (q; q)n−2k ˜ hn−2k,α(x, y|q). (40)

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 41

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Conclusion In this work, (i) we have introduced gdq-H2P ˜ hn,α(x, y|q) and derived several properties. (ii) Also, we have derived implicit summation formula for gdq-H2P ˜ hn,α(x, y|q) by using different analytical means on their generating function. (iii) For y = 1, the assertion of Theorem 2 can be expressed in terms of generalized discrete q-Hermite II polynomials ˜ hn,α(x; q). The assertion of Theorem 2 can be written in terms of discrete q-Hermite II polynomials ˜ hn(x; q) by choosing y = 1 and α = −1/2. (iv) This process can be extend to summation formula for more generalized forms of q-Hermite polynomials. This study is under way.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 42

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Conclusion In this work, (i) we have introduced gdq-H2P ˜ hn,α(x, y|q) and derived several properties. (ii) Also, we have derived implicit summation formula for gdq-H2P ˜ hn,α(x, y|q) by using different analytical means on their generating function. (iii) For y = 1, the assertion of Theorem 2 can be expressed in terms of generalized discrete q-Hermite II polynomials ˜ hn,α(x; q). The assertion of Theorem 2 can be written in terms of discrete q-Hermite II polynomials ˜ hn(x; q) by choosing y = 1 and α = −1/2. (iv) This process can be extend to summation formula for more generalized forms of q-Hermite polynomials. This study is under way.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 43

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Conclusion In this work, (i) we have introduced gdq-H2P ˜ hn,α(x, y|q) and derived several properties. (ii) Also, we have derived implicit summation formula for gdq-H2P ˜ hn,α(x, y|q) by using different analytical means on their generating function. (iii) For y = 1, the assertion of Theorem 2 can be expressed in terms of generalized discrete q-Hermite II polynomials ˜ hn,α(x; q). The assertion of Theorem 2 can be written in terms of discrete q-Hermite II polynomials ˜ hn(x; q) by choosing y = 1 and α = −1/2. (iv) This process can be extend to summation formula for more generalized forms of q-Hermite polynomials. This study is under way.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 44

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Conclusion In this work, (i) we have introduced gdq-H2P ˜ hn,α(x, y|q) and derived several properties. (ii) Also, we have derived implicit summation formula for gdq-H2P ˜ hn,α(x, y|q) by using different analytical means on their generating function. (iii) For y = 1, the assertion of Theorem 2 can be expressed in terms of generalized discrete q-Hermite II polynomials ˜ hn,α(x; q). The assertion of Theorem 2 can be written in terms of discrete q-Hermite II polynomials ˜ hn(x; q) by choosing y = 1 and α = −1/2. (iv) This process can be extend to summation formula for more generalized forms of q-Hermite polynomials. This study is under way.

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials

SLIDE 45

Notations and definitions Generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Connection formulae for the generalized discrete q-Hermite II polynomials {˜ hn,α(x, y|q)}∞ n=0 Conclusion

Thank you for attention ! ! !

Sama Arjika Summation formula for generalized discrete q-Hermite II polynomials