Abstract
Monic families of bivariate Askey–Wilson polynomials and q-Racah polynomials are explicitly given. Monic families of bivariate orthogonal polynomials, both in quadratic and q-quadratic lattices, are also explicitly given using appropriate limit relations.
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1 Introduction
Univariate Askey–Wilson polynomials can be explicitly defined as [13, page 415]
where the q-Pochhammer symbol is defined as
and the basic hypergeometric series is defined as
with
The polynomial \(p_{n}(x;a,b,c,d \vert q)\) is a polynomial of degree n in the q-quadratic lattice [7, 22]
Let us introduce the divided-difference operators \({\mathbb {D}}_{x}\) and \({\mathbb {S}}_{x}\) [20, 21, 31]
The above operators transform polynomials of degree n in the lattice x(s) into polynomials of, respectively, degree \(n-1\) and n in the same variable x(s). Then, univariate Askey–Wilson polynomials satisfy the second-order linear divided-difference equation [9]
where \(\phi _{1}\) is a polynomial of degree two in the lattice x(s) given by
\(\tau _{1}\) is a polynomial of degree one in the lattice x(s)
and
By introducing
it is possible to rewrite the q-difference equation in the form [13, page 418]
where
is a rational function.
Bivariate Askey–Wilson polynomials have been introduced by Gasper and Rahman [14] as
Both families (1.5) and (1.6) are polynomials of total degree \(n+m\) in the variables x(s) and \(y(t)=x(t)\). Iliev [17] obtained a partial divided-difference equation for the Askey–Wilson polynomials defined by Gasper–Rahman (in arbitrary dimension) containing rational coefficients. The equation can be rewritten in terms of a divided-difference equation of hypergeometric type with polynomial coefficients, but it is required to consider a fourth-order equation [28]. Moreover, in [28], a monic family of bivariate Askey–Wilson polynomials is introduced, from the three-term recurrence relations they satisfy, but the explicit expression has not been provided yet in the literature, to the best of our knowledge. Monic bivariate orthogonal polynomials as eigenfunctions of partial differential (difference, q-difference or divided-difference) operators have been analyzed in detail in [1,2,3, 5, 26]. The q-analogues, i.e., bivariate and multivariate orthogonal polynomials on q-quadratic lattices have similar properties in terms of eigenfunctions of the fourth-order divided-difference equations, and representation as products of univariate orthogonal polynomials on q-quadratic lattices.
Let us introduce some other bivariate extensions of classical univariate orthogonal polynomials. Let us first consider the q-Racah polynomials defined for \(n=0,1,\ldots ,N\) by [6, 13, 15]
where the factor \((q^N/c)^{n/ 2}\) were chosen, so that certain symmetry properties of the q-Racah polynomials are satisfied. Note that \(r_n(x;a,b,c,N;q)\) is a polynomial of degree n in the q-quadratic lattice
Gasper and Rahman [15] defined the multivariate q-Racah polynomials from which we deduce the bivariate q-Racah polynomials given by
We would like to notice that \(R_{n,m}(x,y;a_1,a_2,a_3,b,N;q)\) is a polynomial of total degree \(n+m\) in the variables \(\mu (x)\) and \(\nu (y),\) where
As for the quadratic cases, let [13, page 190]
be the univariate Racah polynomials, where \((A)_{n}=A(A+1)\cdots (A+n-1)\) with \((A)_{0}=1\) denotes the Pochhammer symbol. The polynomial \(r_{n}(\alpha ,\beta ,\gamma ,\delta ;s)\) is a polynomial of degree 2n in s and of degree n in the quadratic lattice [7, 22]
Univariate Racah polynomials satisfy the second-order linear divided-difference equation [9]
where \(\phi \) is a polynomial of degree two in the lattice \(\eta (s)\)
\(\tau \) is a polynomial of degree one in the lattice \(\eta (s)\)
and
Equation (1.13) can be also written in many other forms, e.g., [13, Eq. (9.2.5)]
where B(s) and D(s) are the rational functions given by
The extension to the multivariable situation was given by Tratnik in [30] and later analyzed by Geronimo and Iliev in [11]. In the two-dimensional situation, the bivariate Racah polynomials given in [11] are defined in terms of univariate Racah polynomials (1.11) as
which are polynomials in the lattices \(x(s)=s(s+\beta _{1})\) and \(y(t)=t(t+\beta _{2})\). If we consider the substitutions
then the above polynomials exactly coincide with the bivariate Racah polynomials of parameters \(a_{1}\), \(a_{2}\), \(a_{3}\), \(\gamma \), and \(\eta \) introduced by Tratnik [30, Eq. (2.1)]. In [11], an equation for bivariate Racah polynomials is given, which involves 9 rational coefficients. This equation can be rewritten as divided-difference equation of hypergeometric type, but again the equation must be of fourth order [27]. Moreover, as in the q-quadratic case, in [27] a monic family of bivariate Racah polynomials is introduced, from the three-term recurrence relations they satisfy, but again the explicit expression has not been provided.
The main aim of this paper is to give explicitly a representation of bivariate monic Askey–Wilson polynomials—see Sect. 2. Monic bivariate q-Racah polynomials are also explicitly given in Sect. 3. Using appropriate limit relations, families of monic bivariate orthogonal polynomials on q-quadratic lattices are introduced in Sect. 4. Monic bivariate Racah polynomials are explicitly represented in Sect. 5. Finally, some other families of monic bivariate orthogonal polynomials on quadratic lattices are deduced in Sect. 6.
2 Monic Bivariate Askey–Wilson Polynomials
In a very interesting contribution, Iliev [17] obtained a partial divided-difference equation for the Askey–Wilson polynomials defined by Gasper–Rahman (in arbitrary dimension). This equation, containing rational coefficients, can be rewritten as a fourth-order divided-difference equation of hypergeometric type [28].
Theorem 2.1
Let
The bivariate Askey–Wilson polynomials defined in (1.5) are solution of the following fourth-order linear partial divided-difference equation:
where
\(P_{n,m}(s,t)\) stands for \(P_{n,m}(s,t;a,b,c,d,e_2|q)\), with
Our first contribution is to present an explicit monic solution of the above fourth-order linear partial divided-difference equation. In doing so, it is extremely important to choose appropriate bases for both variables s and t. While the first variable involves just a classical quadratic basis \((a q^s,a q^{-s};\,q)_{k}\), the basis for the second variable in the summation contains also the summation index of the first variable, i.e., \((a e_{2} q^{k+t},a e_{2} q^{k-t};\,q)_{p}\).
Theorem 2.2
The fourth-order divided-difference equation (2.1) has the following monic solution:
Proof
Let
Using the definitions of the divided-difference operators as well as the latter definitions, the following relations hold true:
If we apply the fourth-order divided-difference Eqs. (2.1) to (2.2), the result follows using the latter properties. \(\square \)
Remark 1
Let
be the column vector of bivariate Askey–Wilson polynomials, where \(P_{n,m}(s,t;a,b,c,d,e_2|q)\) are defined in (1.5). Let us also introduce the column vector of monic bivariate Askey–Wilson polynomials
where \(\hat{P}_{n,m}(s,t)\) are defined in (2.2). Then, by computing the leading coefficients of the bivariate Askey–Wilson polynomials \(P_{n,m}(s,t;a,b,c,d,e_2|q)\) defined in (1.5), it yields
where the upper triangular matrix \({\mathbb {W}}_{n}\) of size \((n+1)\times (n+1)\) is defined as
3 Monic Bivariate q-Racah Polynomials
Theorem 3.1
The bivariate q-Racah polynomials are solution of a fourth-order linear partial divided-difference equation of the form
where \( R_{n,m}(x,y;a_1,a_2,a_3,b,N;q):= R_{n,m}(x,y;q)\) and
Similarly as in the Askey–Wilson case, it is possible to obtain a explicit expression for the monic bivariate q-Racah polynomials
Theorem 3.2
The fourth-order divided-difference Eq. (3.1) has the following monic solution
Remark 2
Let
be the column vector of bivariate q-Racah polynomials, where \(R_{n,m}(s,t;a_{1},a_{2},a_{3},b,N,q)\) are defined in (1.9). Let us also introduce the column vector of monic bivariate q-Racah polynomials
where \(\hat{R}_{n,m}(s,t)\) are defined in (3.2). Then, by computing the leading coefficients of the bivariate q-Racah polynomials \(R_{n,m}(s,t;a_{1},a_{2},a_{3},b,N,q)\) defined in (1.9), it yields
where the upper triangular matrix \({\mathbb {T}}_{n}\) of size \((n+1)\times (n+1)\) is defined as
4 Limit Transitions on q-Quadratic Lattices
4.1 Bivariate Monic Dual q-Hahn Polynomials
The family of bivariate monic dual q-Hahn polynomials follows by taking the limit as \(b \rightarrow 0\) from (3.2):
4.2 Bivariate Monic q-Hahn Polynomials
If we take the limit \(a_1\rightarrow 0\) and replace \(b\leftarrow a_1\), \(a_2\leftarrow a_2q\), \(a_3\leftarrow a_3q\) in (3.2), this yields the bivariate monic q-Hahn polynomials
4.3 Bivariate Monic q-Krawtchouk Polynomials
If we let \(a_1\rightarrow \infty \) in \(\hat{H}_{n,m}(x,y;q)\) and replace \(a_2\leftarrow a_1\), \(a_3\leftarrow a_2\), we obtain the bivariate monic q-Krawtchouk polynomials
4.4 Bivariate Monic q-Meixner Polynomials
In the bivariate monic q-Racah polynomials, if we substitute \(q^{-N} \leftarrow \beta \), take the limits \(b \rightarrow \infty \) and \(a_1 \rightarrow 0\) and replace \(a_2 \leftarrow a_1\), \(a_3\leftarrow a_2\), we obtain the bivariate monic q-Meixner polynomials given by
4.5 Bivariate Monic q-Charlier Polynomials
The \(\beta \rightarrow 0\) limit of the bivariate monic q-Meixner polynomials gives the bivariate monic q-Charlier polynomials
5 Monic Bivariate Racah Polynomials
Let
In [11], a difference equation for multivariate Racah polynomials was giving, involving rational coefficients. In the bivariate case, the equation was rewritten using appropriate divided-difference operators, in terms of a fourth-order linear partial divided-difference equation with polynomial coefficients in [27].
Theorem 5.1
The bivariate Racah polynomials defined in (1.14) are solution of the following fourth-order linear partial divided-difference equation:
where \(R_{n,m}(s,t):=R_{n,m}(s,t;\beta _{0},\beta _{1},\beta _{2},\beta _3,N)\), and the coefficients \(f_i\), \(i=1,\ldots ,8\) are polynomials in the lattices x(s) and y(t) defined in (5.1) given by
Next, we present the monic solution of the above fourth-order linear partial divided-difference equation. In doing so, as it happens in the q-quadratic case, it is extremely important to choose appropriate bases for both variables s and t. Similarly, the first variable involves just a classical quadratic basis \(\{(-s)_j (s+\beta _{1})_j \}_{j \ge 0}\), and the basis for the second variable in the summation contains also the summation index of the first variable, i.e., \(\{(j-t)_r (j+t+\beta _{2})_r\}_{r \ge 0}\).
Theorem 5.2
The fourth-order divided-difference Eq. (5.2) has the following monic solution:
In the particular case \(n=0\), the above expression reduces to
which can be expressed as
Similarly, in the case \(m=0\), we have
or equivalently
Proof
Let
Using the definitions of the divided-difference operators as well as the latter definitions, the following relations hold true:
If we apply the fourth-order divided-difference Eqs. (5.2)–(5.3), the result follows using the latter properties. \(\square \)
Remark 3
Let
be the column vector of bivariate Racah polynomials, where \(R_{n,m}(s,t;\beta _{0},\beta _{1},\beta _{2},\beta _3,N)\) are defined in (1.14), assuming the substitutions (1.15). Let us also introduce the column vector of monic bivariate Racah polynomials
where \(\hat{P}_{n,m}(s,t)\) are defined in (5.3). Then, by computing the leading coefficients of the bivariate Racah polynomials \(R_{n,m}(s,t;\beta _{0},\beta _{1},\beta _{2},\beta _3,N)\) are defined in (1.14), it yields
where the upper triangular matrix \({\mathbb {U}}_{n}\) of size \((n+1)\times (n+1)\) is defined as
6 Limit Transitions on Quadratic Lattices
6.1 Bivariate Monic Wilson Polynomials
Let [11, p. 443]
Under the above change of variables monic bivariate Racah polynomials (1.14) transform into the monic bivariate Wilson polynomials (in a similar way as in the univariate case [13, p. 196])
6.2 Bivariate Monic Continuous Dual Hahn Polynomials
If we take the limit \(b\rightarrow \infty \) in (6.2) (after redefining \(c\rightarrow b\), \(d\rightarrow c\)), we obtain the monic bivariate continuous dual Hahn polynomials [29]
6.3 Bivariate Monic Continuous Hahn Polynomials
If we take the limit as \(\epsilon \rightarrow \infty \), the resulting polynomials are the bivariate monic continuous Hahn polynomials
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Acknowledgements
The authors would like to thank both reviewers for helpful suggestions that improved a preliminary version of the manuscript. The work of the first author has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grants MTM2016-75140-P and PID2020-113275GB-I00, cofinanced by the European Community fund FEDER.
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Area, I., Tefo, Y.G. Monic Bivariate Polynomials on Quadratic and q-Quadratic Lattices. Mediterr. J. Math. 19, 132 (2022). https://doi.org/10.1007/s00009-022-02049-8
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DOI: https://doi.org/10.1007/s00009-022-02049-8
Keywords
- Bivariate Askey–Wilson polynomials
- monic bivariate Askey–Wilson polynomials
- bivariate Racah polynomials
- monic bivariate Racah polynomials
- partial divided-difference equation
- partial difference equation
- nonuniform lattice
- quadratic lattice
- q-quadratic lattice