Around Poisson-Mehler summation formula

Pawel J. SZABLOWSKİ

Article Details

Around Poisson-Mehler summation formula

Authors:

Pawel J. SZABLOWSKİ

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2016 Vol.45 Issue 6, pp.1729-1742

Document Type:

Article

Publication Date:

11/2016

Document Language:

English

Document ID (Org No.):

2000452394

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X

Abstract:

 

We study polynomials in xxx and yyy of degree n+mn+mn+m: {Qm,n(x,y|t,q)}n,m≥0{Qm,n(x,y|t,q)}n,m≥0\{Q_{m,n}(x,y|t,q)\}_{n,m\geq0} that are related to the generalization of Poisson-Mehler formula i.e. to the expansion ∑i≥0ti[i]q!Hi+n(x|q)Hm+i(y|q)=Qn,m(x,y|t,q)∑i≥0Hi(x|q)Hm(y|q)∑i≥0ti[i]q!Hi+n(x|q)Hm+i(y|q)=Qn,m(x,y|t,q)∑i≥0Hi(x|q)Hm(y|q)\sum_{i\geq0}\dfrac{t^{i}}{[i]_q!}H_{i+n}(x|q)H_{m+i}(y|q)=Q_{n,m}(x,y|t,q)\sum_{i\geq0}H_i(x|q)H_m(y|q), where {Hn(x|q)}n≥−1{Hn(x|q)}n≥−1\{H_n(x|q)\}_{n\geq-1} are the so-called q−q−q-Hermite polynomials (qH). In particular we show that the spaces span{Qi,n−i(x,y|t,q):i=0,⋯,n}n≥0span{Qi,n−i(x,y|t,q):i=0,⋯,n}n≥0span\{Q_{i,n-i}(x,y|t,q):i=0,\cdots,n\}_{n\geq0} are orthogonal with respect to a certain measure (two dimensional (t,q)−(t,q)−(t,q)-Normal distribution) on the square {(x,y):|x|,|y|≤2/√1−q}{(x,y):|x|,|y|≤2/1−q}\{(x,y):|x|,|y|\leq2/\sqrt{1-q}\} being a generalization of two-dimensional Gaussian measure. We study structure of these polynomials showing in particular that they are rational functions of parameters ttt and qqq. We use them in various infinite expansions that can be viewed as simple generalization of the Poisson-Mehler summation formula. Further we use them in the expansion of the reciprocal of the right hand side of the Poisson-Mehler formula. .

Article Details

Around Poisson-Mehler summation formula

Authors: Pawel J. SZABLOWSKİ

Publisher: Hacettepe University

Main Topics: Mathematics

Journal: Hacettepe Journal of Mathematics and Statistics 2016 Vol.45 Issue 6, pp.1729-1742

Document Type: Article

Publication Date: 11/2016

Document Language: English

Document ID (Org No.): 2000452394

Frequency: Bimonthly

eISSN: 2651-477X

pISSN: 2651-477X

Issues

Authors:

Pawel J. SZABLOWSKİ

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2016 Vol.45 Issue 6, pp.1729-1742

Document Type:

Article

Publication Date:

11/2016

Document Language:

English

Document ID (Org No.):

2000452394

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X