Classical orthogonal polynomials are solutions of a second order differential, difference, or q-difference equation for continuous, discrete, or q-discrete variables, respectively. Furthermore, all such systems satisfy a three-term recurrence equation of the form: <Equation ID="Equ34"> <EquationSource Format="TEX">$\begin{aligned} p_{n+1}(x) = (A_n x + B_n)p_n(x) - C_n p_{n-1}(x), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p_{-1}=0,~p_0=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Given a holonomic three-term recurrence equation, we implement in <Emphasis FontCategory="NonProportional">Maxima</Emphasis> and <Emphasis FontCategory="NonProportional">Maple</Emphasis> an algorithm which detects its classical orthogonal polynomial solutions for the continuous, discrete, and q-discrete variables when they exist. With our implementations, the results obtained using the Maple implementations by Koepf and Schmersau (Appl Math Comput 128:303–327, 2002) and Koorwinder and Swarttouw (Priv Commun, 1998) are easily recovered. In addition, we obtain new relations that extend beyond those previously established in the literature.

Recurrence equations and their classical continuous, discrete, and q-discrete orthogonal polynomial solutions

期刊论文
出版模式：

开放获取
发表日期：

2026年4月21日

查看全文

Ngozi Pleasure Nwoku,
Daniel Duviol Tcheutia,
Wolfram Koepf

摘要

Classical orthogonal polynomials are solutions of a second order differential, difference, or q-difference equation for continuous, discrete, or q-discrete variables, respectively. Furthermore, all such systems satisfy a three-term recurrence equation of the form: $\begin{aligned} p_{n+1}(x) = (A_n x + B_n)p_n(x) - C_n p_{n-1}(x), \end{aligned}$ $\begin{matrix} p_{n + 1} (x) = (A_{n} x + B_{n}) p_{n} (x) - C_{n} p_{n - 1} (x), \end{matrix}$ for $n\ge 0$ $n \geq 0$ with $ p_{-1}=0,~p_0=1$ $p_{- 1} = 0, p_{0} = 1$ . Given a holonomic three-term recurrence equation, we implement in Maxima and Maple an algorithm which detects its classical orthogonal polynomial solutions for the continuous, discrete, and q-discrete variables when they exist. With our implementations, the results obtained using the Maple implementations by Koepf and Schmersau (Appl Math Comput 128:303–327, 2002) and Koorwinder and Swarttouw (Priv Commun, 1998) are easily recovered. In addition, we obtain new relations that extend beyond those previously established in the literature.