The big q-Jacobi polynomials, a foundational family within the q-Askey scheme of hypergeometric orthogonal polynomials, play a pivotal role in q-analysis. This paper establishes a necessary and sufficient condition for an analytic function to admit an expansion in terms of big q-Jacobi polynomials. As applications, we derive two novel q-integral formulas, which include several famous integrals as special cases, such as the Andrews-Askey integral and Liu’s q-integral formulas. Moreover, we also discovered some q-identities, for example <Equation ID="Equ45"> <EquationSource Format="TEX">$\begin{aligned}&\frac{1}{(q;q)_{\infty }} \sum _{n=0}^{\infty }\frac{ q^{n^2+2n}}{(q^2,q;q)_n} =\sum _{n=0}^{\infty }\frac{q^n}{(q^2,q;q)_n},\\&\frac{1}{(q;q)_{\infty }} \sum _{n=0}^{\infty }\frac{(-1)^n q^{n^2+n}}{(q^2;q^2)_n} =\sum _{n=0}^{\infty }\frac{q^n}{(q^2;q^2)_n},\\&\frac{1}{(q^2;q^2)_{\infty }} \sum _{n=0}^{\infty }\frac{(-1)^n q^{n^2+2n}}{(q^2;q^2)_n} =\sum _{n=0}^{\infty }\frac{q^{2n}(q;q)_{2n}}{(q^2;q^2)_n^2}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <mfrac> <mn>1</mn> <msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <msup> <mi>q</mi> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mfrac> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <msup> <mi>q</mi> <mi>n</mi> </msup> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mfrac> <mn>1</mn> <msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <msup> <mi>q</mi> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>n</mi> </mrow> </msup> </mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mfrac> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <msup> <mi>q</mi> <mi>n</mi> </msup> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mfrac> <mn>1</mn> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <msup> <mi>q</mi> <mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msub> </mfrac> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <msup> <mi>q</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>;</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> <mn>2</mn> </msubsup> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>

Characterizations of big q-Jacobi polynomials via q-partial differential equations and their applications

期刊论文
发表日期：

2026年5月9日

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Qi Bao,
Jin Liu

摘要

The big q-Jacobi polynomials, a foundational family within the q-Askey scheme of hypergeometric orthogonal polynomials, play a pivotal role in q-analysis. This paper establishes a necessary and sufficient condition for an analytic function to admit an expansion in terms of big q-Jacobi polynomials. As applications, we derive two novel q-integral formulas, which include several famous integrals as special cases, such as the Andrews-Askey integral and Liu’s q-integral formulas. Moreover, we also discovered some q-identities, for example $\begin{aligned}&\frac{1}{(q;q)_{\infty }} \sum _{n=0}^{\infty }\frac{ q^{n^2+2n}}{(q^2,q;q)_n} =\sum _{n=0}^{\infty }\frac{q^n}{(q^2,q;q)_n},\\&\frac{1}{(q;q)_{\infty }} \sum _{n=0}^{\infty }\frac{(-1)^n q^{n^2+n}}{(q^2;q^2)_n} =\sum _{n=0}^{\infty }\frac{q^n}{(q^2;q^2)_n},\\&\frac{1}{(q^2;q^2)_{\infty }} \sum _{n=0}^{\infty }\frac{(-1)^n q^{n^2+2n}}{(q^2;q^2)_n} =\sum _{n=0}^{\infty }\frac{q^{2n}(q;q)_{2n}}{(q^2;q^2)_n^2}. \end{aligned}$ $\begin{matrix} \frac{1}{{(q; q)}_{\infty}} \sum_{n = 0}^{\infty} \frac{q^{n^{2} + 2 n}}{{(q^{2}, q; q)}_{n}} = \sum_{n = 0}^{\infty} \frac{q^{n}}{{(q^{2}, q; q)}_{n}}, \\ \frac{1}{{(q; q)}_{\infty}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} q^{n^{2} + n}}{{(q^{2}; q^{2})}_{n}} = \sum_{n = 0}^{\infty} \frac{q^{n}}{{(q^{2}; q^{2})}_{n}}, \\ \frac{1}{{(q^{2}; q^{2})}_{\infty}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} q^{n^{2} + 2 n}}{{(q^{2}; q^{2})}_{n}} = \sum_{n = 0}^{\infty} \frac{q^{2 n} {(q; q)}_{2 n}}{{(q^{2}; q^{2})}_{n}^{2}} . \end{matrix}$