Little q-Jacobi: Special case
Little q-Jacobi: Special case
Little q-Legendre
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqLegendre{n}@{x}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^1\littleqLegendre{m}@{x}{q}\littleqLegendre{n}@{x}{q}\,d_qx=(1-q)\sum_{k=0}^{\infty}q^k\littleqLegendre{m}@{q^k}{q}\littleqLegendre{n}@{q^k}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^1\littleqLegendre{m}@{x}{q}\littleqLegendre{n}@{x}{q}\,d_qx=\frac{(1-q)q^n}{(1-q^{2n+1})}\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\littleqLegendre{n}@{x}{q}=A_n\littleqLegendre{n+1}@{x}{q}-\left(A_n+C_n\right)\littleqLegendre{n}@{x}{q}+C_n\littleqLegendre{n-1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1-q^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\moniclittleqLegendre{n}@@{x}{q}=\moniclittleqLegendre{n+1}@@{x}{q}+(A_n+C_n)\moniclittleqLegendre{n}@@{x}{q}+A_{n-1}C_n\moniclittleqLegendre{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1-q^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqLegendre{n}@{x}{q}=\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\moniclittleqLegendre{n}@@{x}{q} }}
q-Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)(1-q^{n+1})xy(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=qx-1}}
&
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqLegendre{n}@{x}{q}=\frac{q^{\binomial{n}{2}}(1-q)^n}{\qPochhammer{q}{q}{n}} \left(\mathcal{D}_{q^{-1}}\right)^n\left[\qPochhammer{qx}{q}{n}x^n\right] }}
Generating function
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{0}{1}@@{-}{q}{q}{qxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{q}{q}{xt}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q,q}{q}{n}}\littleqLegendre{n}@{x}{q}t^n }}
Limit relation
Little q-Legendre polynomial to Legendre / Spherical polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\littleqLegendre{n}@{x}{q}=\LegendrePoly{n}@{1-2x} }}