View source for Little q-Jacobi: Special case

{{DISPLAYTITLE:Little ''q''-Jacobi: Special case}}
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== Little ''q''-Jacobi: Special case ==

== Little ''q''-Legendre ==

== Basic hypergeometric representation ==

<math id="KLS:14.12:28">{\displaystyle 
\littleqLegendre{n}@{x}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx}
}</math>

== Orthogonality relation(s) ==

<math id="KLS:14.12:29">{\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}
}</math><br />
<math id="KLS:14.12:30">{\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}
}</math>

== Recurrence relation ==

<math id="KLS:14.12:31">{\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}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle C_n=q^n\frac{(1-q^n)}{(1+q^n)(1-q^{2n+1})}}</math> &<br />
<math>{\displaystyle A_n=q^n\frac{(1-q^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}</math></div><br />

== Monic recurrence relation ==

<math id="KLS:14.12:34">{\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}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle C_n=q^n\frac{(1-q^n)}{(1+q^n)(1-q^{2n+1})}}</math> &<br />
<math>{\displaystyle A_n=q^n\frac{(1-q^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}</math></div><br />
<math id="KLS:14.12:35">{\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}
}</math>

== ''q''-Difference equation ==

<math id="KLS:14.12:36">{\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)
}</math>
<div align="right">Substitution(s): <math>{\displaystyle D(x)=x-1}</math> &<br />
<math>{\displaystyle B(x)=qx-1}</math> &<br />
<math>{\displaystyle y(x)=\littleqLegendre{n}@{x}{q}}</math></div><br />

== Rodrigues-type formula ==

<math id="KLS:14.12:40">{\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]
}</math>

== Generating function ==

<math id="KLS:14.12:41">{\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
}</math>

== Limit relation ==

=== Little ''q''-Legendre polynomial to Legendre / Spherical polynomial ===

<math id="KLS:14.12:42">{\displaystyle 
\lim_{q\rightarrow 1}\littleqLegendre{n}@{x}{q}=\LegendrePoly{n}@{1-2x}
}</math>
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