Next Article in Journal
Study of Lotka–Volterra Biological or Chemical Oscillator Problem Using the Normalization Technique: Prediction of Time and Concentrations
Next Article in Special Issue
On Multivariate Bernstein Polynomials
Previous Article in Journal
Emotional Intelligence of Engineering Students as Basis for More Successful Learning Process for Industry 4.0
Previous Article in Special Issue
New Stability Criteria for Discrete Linear Systems Based on Orthogonal Polynomials

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials

by 3,* and
1
Department of Mathematics, K. N. Toosi University of Technology, Tehran P.O. Box 16315-1618, Iran
2
School of Mathematical and Computational Science, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE C1A 4P3, Canada
3
Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(8), 1323; https://doi.org/10.3390/math8081323
Received: 20 July 2020 / Revised: 4 August 2020 / Accepted: 5 August 2020 / Published: 8 August 2020

## Abstract

:
Using the Sturm–Liouville theory in q-difference spaces, we prove the finite orthogonality of q-Pseudo Jacobi polynomials. Their norm square values are then explicitly computed by means of the Favard theorem.

## 1. Introduction

For $α , β > − 1$, the Jacobi polynomials are defined as [1]
$P n ( α , β ) ( x ) = ( − 1 ) n 2 n n ! ( 1 − x ) − α ( 1 + x ) − β d n d x n ( 1 − x ) α ( 1 + x ) β ( 1 − x 2 ) n .$
Another representation of Jacobi polynomials is as [2,3]
$P n ( α , β ) ( x ) = ( α + 1 ) n n ! 2 F 1 − n , n + α + β + 1 α + 1 | 1 − x 2 = ( α + 1 ) n n ! ∑ k = 0 n ( − n ) k ( α + β + 1 ) k ( α + 1 ) k ( 1 − x ) k 2 k k ! ,$
where
$( a ) k : = ∏ j = 0 k − 1 ( a + j ) , ( a ) 0 : = 1 ,$
and
$r F s a 1 , ⋯ , a r b 1 , ⋯ , b s | z : = ∑ k = 0 ∞ ( a 1 ) k ⋯ ( a r ) k ( b 1 ) k ⋯ ( b s ) k z k k ! ,$
in which $a 1 , a 2 , ⋯ , a r , b 1 , b 2 , ⋯ , b s , z ∈ C$ and $b 1 , ⋯ , b s ≠ 0 , − 1 , − 2 , ⋯ , − ( k − 1 )$.
The weight function corresponding to Jacobi polynomials is known in statistics as the shifted beta distribution
$w ( x ; α , β ) = ( 1 − x ) α ( 1 + x ) β , x ∈ [ − 1 , 1 ] .$
An interesting subclass of Jacobi polynomials is when $α = − u + i v$ and $β = − u − i v$ for $i 2 = − 1$ in (2), so that the real polynomials
$J n ( u , v ) ( x ) = ( − i ) n P n ( − u + i v , − u − i v ) ( i x ) ,$
satisfy the equation
$( 1 + x 2 ) J n ″ ( x ) + 2 ( ( 1 − u ) x + v ) J n ′ ( x ) − n ( n − 2 u + 1 ) J n ( x ) = 0 .$
It is proved in [4] that ${ J n ( u , v ) ( x ) }$ are finitely orthogonal with respect to the weight function
$w ( x ; u , v ) = ( 1 + x 2 ) − u exp ( 2 v arctan x ) ,$
on $( − ∞ , ∞ )$ and can be explicitly represented in form of hypergeometric functions as
$J n ( u , v ) ( x ) = ( − 2 i ) n ( 1 − u + i v ) n ( n − 2 u + 1 ) n 2 F 1 − n , n − 2 u + 1 1 − u + i v 1 − i x 2 .$
The so-called q-polynomials have found many applications in Eulerian series and continued fractions [3], q-algebras and quantum groups [5,6,7], and q-oscillators [8,9,10]. See also [11,12] in this regard.
It has been acknowledged that the theory of q-special functions is essentially based on the relation
$lim q → 1 1 − q α 1 − q = α .$
Hence, a basic number in q-calculus is defined as
$[ α ] q = 1 − q α 1 − q .$
There is a q-analogue of the Pochhammer symbol (3) (called q-shifted factorial) as
$( a ; q ) k : = ∏ j = 0 k − 1 ( 1 − a q j ) , ( a ; q ) 0 : = 1 .$
Moreover we have
$( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 − a q k ) for 0 < | q | < 1 ,$
and
$( a 1 , a 2 , … , a m ; q ) ∞ = ( a 1 ; q ) ∞ ( a 2 ; q ) ∞ … ( a m ; q ) ∞ .$
There exist several q-analogues of classical hypergeometric orthogonal polynomials that are known as basic hypergeometric orthogonal polynomials [3].
In the present work, using the Sturm–Liouville theory in q-difference spaces, we prove that a special case of big q-Jacobi polynomials is finitely orthogonal on $( − ∞ , ∞ )$. The big q-Jacobi polynomials are defined as
$P n ( x ; a , b , c ; q ) = 3 ϕ 2 q − n , a b q n + 1 , x a q , c q | q ; q ,$
where
$r ϕ s a 1 , ⋯ , a r b 1 , ⋯ , b s | q ; z : = ∑ k = 0 ∞ ( a 1 ; q ) k ⋯ ( a r ; q ) k ( b 1 ; q ) k ⋯ ( b s ; q ) k z k ( q ; q ) k ( − 1 ) k q k ( k − 1 ) 2 1 + s − r ,$
is known as the basic hypergeometric series.
Again, $a 1 , a 2 , ⋯ , a r , b 1 , b 2 , ⋯ , b s , z ∈ C$ and $b 1 , b 2 , ⋯ , b s ≠ 1 , q − 1 , q − 2 , ⋯ , q 1 − k$.
Notice that [3] (p. 15)
$lim q → 1 r ϕ s q a 1 , ⋯ , q a r q b 1 , ⋯ , q b s | q ; ( q − 1 ) 1 + s − r z = r F s a 1 , ⋯ , a r b 1 , ⋯ , b s | z .$
On the other side, if we set $c = 0$, $a = q α$ and $b = q β$ in (8) and then let $q → 1$, we find the Jacobi polynomials (2) as
$lim q → 1 P n ( x ; q α , q β , 0 ; q ) = P n ( α , β ) ( 2 x − 1 ) P n ( α , β ) ( 1 ) .$
Moreover, by referring to (8), one can define another family of big q-Jacobi polynomials [13] with four free parameters as
$P n * ( x ; a , b , c , d ; q ) = P n ( q a c − 1 x ; a , b , − a c − 1 d ; q ) = 3 ϕ 2 q − n , a b q n + 1 , q a c − 1 x a q , − q a c − 1 d | q ; q ,$
which yields
$P n ( x ; a , b , c ; q ) = P n * ( − q − 1 c − 1 x ; a , b , − a c − 1 , 1 ; q ) .$
Because a particular case of Jacobi polynomials (5) are called the pseudo Jacobi polynomials, it is reasonable to similarly consider a special case of big q-Jacobi polynomials preserving the limit relation as $q → 1$. This means that the q-pseudo Jacobi polynomials will be derived by substituting
$a = i q 1 2 ( u − i v ) , b = − i q 1 2 ( u + i v ) , c = i q 1 2 ( − u + i v ) and d = − i q 1 2 ( − u − i v )$
in a special case of the polynomials (8) as
$P n ( c x ; c / b , d / a , c / a ; q ) where a , b , c , d ∈ C and ( a b ) / ( q c d ) > 0 ,$
so that
$lim q → 1 P n ( i q 1 2 ( − u + i v ) x ; − q − u , − q − u , q − u + i v ; q ) = J n ( u , v ) ( x ) J n ( u , v ) ( i ) .$
Therefore, the q-pseudo Jacobi polynomials are defined as
$J n ( u , v ) ( x ; q ) = P n ( i q 1 2 ( − u + i v ) x ; − q − u , − q − u , q − u + i v ; q ) = 3 ϕ 2 q − n , q u + n + 1 , − q 1 + u − i v x − q 1 + 1 2 ( u − i v ) , i q 1 + 1 2 ( u − 3 i v ) | q ; q .$
The main aim of this paper is to apply a q-Sturm–Liouville theorem in order to obtain a finite orthogonality for the real polynomials (11) on $( − ∞ , ∞ )$, which is a new contribution in the literature.
A regular Sturm–Liouville problem of continuous type is a boundary value problem of the form
$d d x K ( x ) d y n ( x ) d x + λ n w ( x ) y n ( x ) = 0 , ( K ( x ) > 0 , w ( x ) > 0 ) ,$
which is defined on an open interval, say $( γ 1 , γ 2 )$ with the boundary conditions
$α 1 y ( γ 1 ) + β 1 y ′ ( γ 1 ) = 0 and α 2 y ( γ 2 ) + β 2 y ′ ( γ 2 ) = 0 ,$
where $α 1 , α 2$ and $β 1 , β 2$ are constant numbers and $K ( x ) ,$ and $w ( x )$ in (12) are to be assumed continuous functions for $x ∈ [ γ 1 , γ 2 ]$. The function $w ( x )$ is called the weight or density function.
Let $y n$ and $y m$ be two eigenfunctions of Equation (12). According to the Sturm–Liouville theory [14], they have an orthogonality property with respect to the weight function $w ( x )$ under the given condition (13), so that we have
$∫ γ 1 γ 2 w ( x ) y n ( x ) y m ( x ) d x = ∫ γ 1 γ 2 w ( x ) y n 2 ( x ) d x ) δ m , n ,$
in which
$δ m , n = 0 ( n ≠ m ) , 1 ( n = m ) .$
There are generally two types of orthogonality for relation (14), i.e. infinitely orthogonality and finitely orthogonality. In the finite case, one has to impose some constraints on n, while in the infinite case, n is free up to infinity [4].
By referring to the differential Equation (6), it is proved in [4] that
$∫ − ∞ ∞ ( 1 + x 2 ) − u exp ( 2 v arctan x ) J n ( u , v ) ( x ) J m ( u , v ) ( x ) d x = 2 π n ! 2 2 n + 1 − 2 u Γ ( 2 u − n ) ( 2 u − 2 n − 1 ) Γ ( u − n + i v ) Γ ( u − n − i v ) δ m , n ⇔ m , n = 0 , 1 , 2 , ⋯ , N = max { m , n } < u − 1 2 and v ∈ R ,$
where $Γ ( . )$ is the well-known gamma function.
Similarly, q-orthogonal functions can be solutions of a q-Sturm-Liouville problem in the form [15]
$D q K ( x ; q ) D q y n ( x ; q ) + λ n , q w ( x ; q ) y n ( x ; q ) = 0 , ( K ( x ; q ) > 0 , w ( x ; q ) > 0 ) ,$
where
$D q f ( x ) = f ( q x ) − f ( x ) ( q − 1 ) x ( x ≠ 0 , q ≠ 1 ) ,$
and (15) satisfies a set of boundary conditions like (13). This means that if $y n ( x ; q )$ and $y m ( x ; q )$ are two eigenfunctions of the q-difference Equation (15), they are orthogonal with respect to a weight function $w ( x ; q )$ on a discrete set [16].
Let $φ ( x )$ and $ψ ( x )$ be two polynomials of degree at most 2 and 1, respectively, as
$φ ( x ) = a x 2 + b x + c and ψ ( x ) = d x + e ( a , b , c , d , e ∈ C , d ≠ 0 ) .$
If ${ y n ( x ; q ) } n$ is a sequence of polynomials that satisfies the q-difference equation [3]
$φ ( x ) D q 2 y n ( x ; q ) + ψ ( x ) D q y n ( x ; q ) + λ n , q y n ( q x ; q ) = 0 ,$
where
$D q 2 ( f ( x ) ) = f ( q 2 x ) − ( 1 + q ) f ( q x ) + q f ( x ) q ( q − 1 ) 2 x 2 ,$
$λ n , q ∈ C$ and $q ∈ R \ { − 1 , 0 , 1 }$, then the following orthogonality relation holds
$∫ ρ 1 ρ 2 w ( x ; q ) y n ( x ; q ) y m ( x ; q ) d q x = ∫ ρ 1 ρ 2 w ( x ; q ) y n 2 ( x ; q ) d q x δ n , m ,$
in which
$∫ ρ 1 ρ 2 f ( t ) d q t = ( 1 − q ) ∑ j = 0 ∞ q j ρ 2 f ( q j ρ 2 ) − ρ 1 f ( q j ρ 1 ) ,$
and $w ( x ; q )$ is a solution of the Pearson q-difference equation
$D q w ( x ; q ) φ ( q − 1 x ) = w ( q x ; q ) ψ ( x ) .$
Note that $w ( x ; q )$ is assumed to be positive and $w ( q − 1 x ; q ) φ ( q − 2 x ) x k$ for $k ∈ N$ must vanish at $x = ρ 1 , ρ 2$.
If $P ¯ n ( x ) = x n + ⋯$ is a monic solution of Equation (16), the eigenvalue $λ n , q$ is explicitly derived as
$λ n , q = − [ n ] q q n ( a [ n − 1 ] q + d ) .$
The q-integral as an inverse of the q-difference operator [3,17,18] is defined as
$∫ 0 x f ( t ) d q t = ( 1 − q ) x ∑ j = 0 ∞ q j f ( q j x ) ( x ∈ R )$
provided that the series converges absolutely. Furthermore, we have
$∫ 0 ∞ f ( t ) d q t = ( 1 − q ) ∑ n = − ∞ ∞ q n f ( q n ) ,$
and
$∫ − ∞ ∞ f ( t ) d q t = ( 1 − q ) ∑ n = − ∞ ∞ q n f ( q n ) + f ( − q n ) .$

## 2. Finite Orthogonality of q-Pseudo Jacobi Polynomials

Let us consider the following q-difference equation
$( q 2 − u x 2 + 2 sin v 2 ln q x + 1 ) D q 2 y n ( x ; q ) + q u − q 2 − u 1 − q x − 2 sin ( v 2 ln q ) ( q 1 − u 2 − q u 2 ) D q y n ( x ; q ) + λ n , q * y n ( q x ; q ) = 0 ,$
with
$λ n , q * = − [ n ] q q n q 2 − u [ n − 1 ] q + q u − q 2 − u 1 − q ,$
for $n = 0 , 1 , 2 , ⋯$ and $q ∈ R \ { − 1 , 0 , 1 }$.
It is clear that
$lim q → 1 λ n , q * = − n ( n − 2 u + 1 ) ,$
gives the same eigenvalues as in the continuous case (6).
Theorem 1.
Let ${ J n ( u , v ) ( x ; q ) } n$ defined in (11) be a sequence of polynomials that satisfies the q-difference Equation (18). Subsequently, we have
$∫ − ∞ ∞ w ( u , v ) ( x ; q ) J n ( u , v ) ( x ; q ) J m ( u , v ) ( x ; q ) d q x = ∫ − ∞ ∞ w ( u , v ) ( x ; q ) J n ( u , v ) ( x ; q ) 2 d q x δ n , m ,$
where $N < u − 1 2$ for $N = max { m , n }$ and the positive function $w ( u , v ) ( x ; q )$ is a solution of the Pearson-type q-difference equation
$D q w ( u , v ) ( x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 = q u − q 2 − u 1 − q x − 2 sin ( v 2 ln q ) ( q 1 − u 2 − q u 2 ) w ( u , v ) ( q x ; q ) ,$
which is equivalent to
$w ( u , v ) ( x ; q ) w ( u , v ) ( q x ; q ) = q u x 2 − 2 q u 2 sin ( v 2 ln q ) x + 1 q − u x 2 + 2 q − u 2 sin ( v 2 ln q ) x + 1 .$
Proof.
First, according to [3] and referring to (7) it is not difficult to verify that
$w ( u , v ) ( x ; q ) = ( i q ( u − i v ) / 2 x , − i q ( u + i v ) / 2 x ; q ) ∞ ( i q ( − u + i v ) / 2 x , − i q ( − u − i v ) / 2 x ; q ) ∞ = x − 2 u ( − i q ( − u + i v ) / 2 x − 1 , i q ( − u − i v ) / 2 x − 1 ; q − 1 ) ∞ ( − i q ( u − i v ) / 2 x − 1 , i q ( u + i v ) / 2 x − 1 ; q − 1 ) ∞ ,$
is a solution of Equation (19).
Now, if Equation (18) is written in the self-adjoint form
$D q w ( u , v ) ( x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J n ( u , v ) ( x ; q ) + λ n , q * w ( u , v ) ( q x ; q ) J n ( u , v ) ( q x ; q ) = 0 ,$
and for m as
$D q w ( u , v ) ( x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J m ( u , v ) ( x ; q ) + λ m , q * w ( u , v ) ( q x ; q ) J m ( u , v ) ( q x ; q ) = 0 ,$
by multiplying (21) by $J m ( u , v ) ( q x ; q )$ and (22) by $J n ( u , v ) ( q x ; q )$ and subtracting each other we get
$( λ m , q * − λ n , q * ) w ( u , v ) ( x ; q ) J m ( u , v ) ( x ; q ) J n ( u , v ) ( x ; q ) = q 2 D q w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J n ( u , v ) ( q − 1 x ; q ) J m ( u , v ) ( x ; q ) − q 2 D q w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J m ( u , v ) ( q − 1 x ; q ) J n ( u , v ) ( x ; q ) .$
Hence, q-integration by parts on both sides of (23) over $( − ∞ , ∞ )$ yields
$( λ m , q * − λ n , q * ) ∫ − ∞ ∞ w ( u , v ) ( x ; q ) J m ( u , v ) ( x ; q ) J n ( u , v ) ( x ; q ) d q x = q 2 ∫ − ∞ ∞ { D q w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J n ( u , v ) ( q − 1 x ; q ) J m ( u , v ) ( x ; q ) − D q w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 D q J m ( u , v ) ( q − 1 x ; q ) J n ( u , v ) ( x ; q ) } d q x = q 2 [ w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 × D q J n ( u , v ) ( q − 1 x ; q ) J m ( u , v ) ( x ; q ) − D q J m ( u , v ) ( q − 1 x ; q ) J n ( u , v ) ( x ; q ) ] − ∞ ∞ .$
Because
$max deg { D q J n ( u , v ) ( q − 1 x ; q ) J m ( u , v ) ( x ; q ) − D q J m ( u , v ) ( q − 1 x ; q ) J n ( u , v ) ( x ; q ) } = m + n − 1 ,$
the left-hand side of (24) is zero if
$lim x → ∞ w ( u , v ) ( q − 1 x ; q ) q 2 − u x 2 + 2 sin ( v 2 ln q ) x + 1 x m + n − 1 = 0 .$
By taking $max { m , n } = N$, relation (25) would be equivalent to
$lim x → ∞ ( − i q ( − u + i v ) / 2 x − 1 , i q ( − u − i v ) / 2 x − 1 ; q − 1 ) ∞ ( − i q ( u − i v ) / 2 x − 1 , i q ( u + i v ) / 2 x − 1 ; q − 1 ) ∞ x 2 N − 2 u + 1 = 0 .$
Note that (26) is valid if and only if
$2 N + 1 − 2 u < 0 or N < u − 1 2 .$
Therefore, the right-hand side of (24) tends to zero and
$∫ − ∞ ∞ w ( u , v ) ( x ; q ) J m ( u , v ) ( x ; q ) J n ( u , v ) ( x ; q ) d q x = 0 ,$
if and only if $m ≠ n$ and $N < u − 1 2$ for $N = max { m , n }$. □
Corollary 1.
The finite polynomial set ${ J n ( u , v ) ( x ; q ) } n = 0 N < u − 1 2$ is orthogonal with respect to the weight function (20) on $( − ∞ , ∞ )$.

#### Computing the Norm Square Value

According to (17), because $J n ( u , v ) ( x ; q )$ is a particular case of the big q-Jacobi polynomials, it satisfies the recurrence relation [3]
$J ¯ n + 1 ( u , v ) ( x ; q ) = x − c n ( u , v ; q ) J ¯ n ( u , v ) ( x ; q ) − d n ( u , v ; q ) J ¯ n − 1 ( u , v ) ( x ; q ) ,$
with the initial terms
$J ¯ 0 ( u , v ) ( x ; q ) = 1 , J ¯ 1 ( u , v ) ( x ; q ) = x + 2 sin ( v 2 ln q ) ( 1 − q ) ( q 2 − u / 2 + q 1 + u / 2 ) ( q u − q 2 − u ) ,$
where
$c n ( u , v ; q ) = 2 sin ( v 2 ln q ) q n ( q u − q 2 n − 2 ) ( q u − q 2 n ) × { ( q u − q n − 1 ) q − u / 2 [ n ] q ( 1 + q ) + ( q 2 − u / 2 + q 1 + u / 2 ) − q n + 1 − u ( 1 − q n + 1 ) ( q 1 − u / 2 + q u / 2 ) } , and d n ( u , v ; q ) = ( q n + 1 − q 2 n + 1 ) ( q u − q n − u ) ( 1 − q ) 2 ( q u − q 2 n − u − 1 ) ( q u − q 2 n − u ) 2 ( q u − q 2 n − u + 1 ) × { 4 sin 2 ( v 2 ln q ) q n − 1 − u / 2 ( 1 − q ) 1 + q − q 2 + q u − q 1 + u − q n − 1 1 − q n − u + 1 ( 1 + q − q 2 ) − q n + 1 ( 1 − q ) − ( q 4 n − 2 u + 2 q 2 n + q 2 u ) } .$
Now, by applying the Favard theorem [19] for the monic type of polynomials (11), we get
$∫ − ∞ ∞ w ( u , v ) ( x ; q ) J ¯ m ( u , v ) ( x ; q ) J ¯ n ( u , v ) ( x ; q ) d q x = μ 0 ∏ k = 1 n d k ( u , v ; q ) δ n , m ,$
where
$μ 0 = ∫ − ∞ ∞ ( i q ( u − i v ) / 2 x , − i q ( u + i v ) / 2 x ; q ) ∞ ( i q ( − u + i v ) / 2 x , − i q ( − u − i v ) / 2 x ; q ) ∞ d q x .$
Hence, it remains to explicitly compute the above $μ 0$. For this purpose, we can refer to the general formula ([13] Formula 128)
$∫ z − q Z ∪ z + q Z ( a x , b x ; q ) ∞ ( c x , d x ; q ) ∞ d q x = ( q , a / c , a / d , b / c , b / d ; q ) ∞ ( a b / ( q c d ) ; q ) ∞ θ ( z − / z + ; q ) θ ( c d z − z + ; q ) θ ( c z − ; q ) θ ( d z − ; q ) θ ( c z + ; q ) θ ( d z + ; q ) ,$
in which
$θ ( x ; q ) = ( x , q / x ; q ) ∞ .$
Therefore, it is enough to replace $z − = − 1 , z + = 1$ in (27) to finally obtain
$μ 0 = ( q , q u − i v , − q u , − q u , q u + i v ; q ) ∞ ( q 2 u − 1 ; q ) ∞ × ( − 1 , − q , − q u , − q u + 1 ; q ) ∞ ( − i q − u + i v 2 , i q − u + i v 2 , − i q − u − i v 2 , i q − u − i v 2 , − i q 1 − − u + i v 2 , i q 1 − − u + i v 2 , − i q 1 − − u − i v 2 , i q 1 − − u − i v 2 ; q ) ∞ .$
For example, the set ${ J n ( 21 , 1 ) ( x ; q ) } n = 0 20$ is a finite sequence of q-orthogonal polynomials that satisfies the orthogonality relation
$∫ − ∞ ∞ ( i q ( 21 − i ) / 2 x , − i q ( 21 + i ) / 2 x ; q ) ∞ ( i q − ( 21 − i ) / 2 x , − i q − ( 21 + i ) / 2 x ; q ) ∞ J ¯ m ( 21 , 1 ) ( x ; q ) J ¯ n ( 21 , 1 ) ( x ; q ) d q x = ( q , q 21 − i , − q 21 , − q 21 , q 21 + i , − 1 , − q , − q 21 , − q 22 ; q ) ∞ ( q 41 , − i q − 21 + i 2 , i q − 21 + i 2 , − i q − 21 − i 2 , i q − 21 − i 2 , − i q 23 − i 2 , i q 23 − i 2 , − i q 23 + i 2 , i q 23 + i 2 ; q ) ∞ ∏ k = 1 n d k ( 21 , 1 ; q ) δ m , n ⟺ m , n < 20 ,$
where
$d k ( 21 , 1 ; q ) = ( q k + 1 − q 2 k + 1 ) ( q 21 − q k − 21 ) ( 1 − q ) 2 ( q 21 − q 2 k − 22 ) ( q 21 − q 2 k − 21 ) 2 ( q 21 − q 2 k − 20 ) × { 4 sin 2 ( 1 2 ln q ) q k − 23 / 2 ( 1 − q ) 1 + q − q 2 + q 21 − q 22 − q k − 1 1 − q k − 20 ( 1 + q − q 2 ) − q k + 1 ( 1 − q ) − ( q 4 k − 42 + 2 q 2 k + q 42 ) } .$

## Author Contributions

Investigation, M.M.-J. and F.S.; validation, M.M.-J., N.S., W.K. and F.S.; conceptualization, M.M.-J. and F.S.; methodology, M.M.-J., N.S., W.K. and F.S.; formal analysis, M.M.-J., N.S., W.K. and F.S.; funding acquisition, W.K; writing—review and editing, M.M.-J., N.S., W.K. and F.S.; writing—original draft preparation, M.M.-J. and F.S. The authors contributed equally to the work. All authors have read and agreed to the published version of the manuscript.

## Funding

The work of the first author has been supported by the Alexander von Humboldt Foundation under the grant number: Ref 3.4—IRN—1128637—GF-E. Partial financial support of this work, under Grant No. GP249507 from the Natural Sciences and Engineering Research Council of Canada, is gratefully acknowledged by the second author.

## Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions for improving the quality of the paper.

## Conflicts of Interest

The authors declare that they have no competing interests.

## References

1. Szegö, G. Orthogonal Polynomials, 4th ed.; Colloquium Publications. XXIII; American Mathematical Society: Providence, RI, USA, 1939. [Google Scholar]
2. Gasper, G.; Rahman, M. Basic Hypergeometric Series, 2nd ed.; Volume 96 of Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
3. Koekoek, R.; Lesky, P.A.; Swarttouw, R.F. Hypergeometric Orthogonal Polynomials and Their q-Analogues; Springer Monographs in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
4. Masjed-Jamei, M. Classical orthogonal polynomials with weight function ((ax + b)2 + (cx + d)2)pexp(qarctan((ax + b)/(cx + d))), x∈(−,) and a generalization of T and F distributions. Integral Transform. Spec. Funct. 2004, 15, 137–153. [Google Scholar] [CrossRef]
5. Vilenkin, N.J.; Klimyk, A.U. Representation of Lie Groups and Special Functions; Volume 75 of Mathematics and its Applications (Soviet Series); Kluwer Academic Publishers Group: Dordrecht, The Netherlands, 1992; Volume 3. [Google Scholar]
6. Koornwinder, T.H. Orthogonal polynomials in connection with quantum groups. In Orthogonal Polynomials (Columbus, OH, 1989); Volume 294 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.; Kluwer Acad. Publ.: Dordrecht, The Netherlands, 1990; pp. 257–292. [Google Scholar]
7. Koornwinder, T.H. Compact quantum groups and q-special functions. In Representations of Lie Groups and Quantum Groups (Trento, 1993); Volume 311 of Pitman Res. Notes Math. Ser.; Longman Sci. Tech.: Harlow, UK, 1994; pp. 46–128. [Google Scholar]
8. Álvarez-Nodarse, R.; Atakishiyev, N.M.; Costas-Santos, R.S. Factorization of the hypergeometric-type difference equation on non-uniform lattices: Dynamical algebra. J. Phys. A 2005, 38, 153–174. [Google Scholar] [CrossRef]
9. Grünbaum, F.A. Discrete models of the harmonic oscillator and a discrete analogue of Gauss’ hypergeometric equation. Ramanujan J. 2001, 5, 263–270. [Google Scholar] [CrossRef]
10. Atakishiyev, N.M.; Klimyk, A.U.; Wolf, K.B. A discrete quantum model of the harmonic oscillator. J. Phys. A 2008, 41, 085201. [Google Scholar] [CrossRef][Green Version]
11. Masjed-Jamei, M.; Soleyman, F.; Koepf, W. Two finite sequences of symmetric q-orthogonal polynomials generated by two q-sturm-liouville problems. Rep. Math. Phys. 2020, 85, 41–55. [Google Scholar] [CrossRef]
12. Masjed-Jamei, M. A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm-Liouville problems. Integral Transform. Spec. Funct. 2007, 18, 871–883. [Google Scholar] [CrossRef][Green Version]
13. Koornwinder, T.H. Additions to the formula lists in “Hypergeometric orthogonal polynomials and their q-analogues” by Koekoek, Lesky and Swarttouw. arXiv 2015, arXiv:1401.0815. [Google Scholar]
14. Nikiforov, A.F.; Uvarov, V.B. Polynomial solutions of hypergeometric type difference equations and their classification. Integral Transform. Spec. Funct. 1993, 1, 223–249. [Google Scholar] [CrossRef]
15. Jirari, A. Second-order Sturm-Liouville difference equations and orthogonal polynomials. Mem. Am. Math. Soc. 1995, 116, 542. [Google Scholar] [CrossRef]
16. Nikiforov, A.F.; Suslov, S.K.; Uvarov, V.B. Classical Orthogonal Polynomials of a Discrete Variable; Springer Series in Computational Physics; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
17. Jackson, F.H. On q-definite integrals. Q. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
18. Thomae, J. Beiträge zur Theorie der durch die Heinesche Reihe darstellbaren Functionen. J. Reine Angew. Math. 1869, 70, 258–281. [Google Scholar]
19. Chihara, T.S. An Introduction to Orthogonal Polynomials; Gordon and Breach Science Publishers: New York, NY, USA, 1978. [Google Scholar]

## Share and Cite

MDPI and ACS Style

Masjed-Jamei, M.; Saad, N.; Koepf, W.; Soleyman, F. On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials. Mathematics 2020, 8, 1323. https://doi.org/10.3390/math8081323

AMA Style

Masjed-Jamei M, Saad N, Koepf W, Soleyman F. On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials. Mathematics. 2020; 8(8):1323. https://doi.org/10.3390/math8081323

Chicago/Turabian Style

Masjed-Jamei, Mohammad, Nasser Saad, Wolfram Koepf, and Fatemeh Soleyman. 2020. "On the Finite Orthogonality of q-Pseudo-Jacobi Polynomials" Mathematics 8, no. 8: 1323. https://doi.org/10.3390/math8081323

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.