.
Big q-Legendre polynomials
In mathematics, the big q-Legendre polynomials, are orthogonal polynomials defined with basic hypergeometric function as[1]
\( \displaystyle P_n(x;c;q)={}_3\phi_2(q^{-n},q^{n+1},x;q,cq;q,q) \)
Orthogonality relation
\( \int_{cq}^q P_m(x;c;q)P_n(x;c;q) \, d_qx=q(1-c)\frac{1-q}{1-q^{2n+1}}\frac{(c^{-1}q;q)_n}{(cq;q)_n}(-cq^2)^n q^{n \choose 2}\delta_{mn} \)
Limiting relations
Big q-Legendre polynomials→Legendre polynomials
\( \displaystyle\lim_{q \to 1} P_n(x;0;q)=P_n(2x-1) \)
Check with 7th-order big q-Legendre polynomials
From definition of big q-Legendre polynomials, we have
\( \begin{align} & qL=P_7(x;0;q) \\ = {} & 1+\frac {q}{(1-q)^2}-\frac {qx}{(1-q)^2} -\frac {q^9}{(1-q)^2} + \frac {xq^9}{(1-q)^2} -\frac {1}{q^6 (1-q)^2} \\[6pt] & {} + \frac {x}{q^6 (1-q)^2} + \frac {q^2}{(1-q)^2} -\frac {xq^2}{(1-q)^2} \\[6pt] & {} + (1-q^{-7}) ( 1-q^{-6}) (1-q^8) (1-q^9) (1-x) (1-qx) q^2 (1-q)^{-2} (1-q^2)^{-2} \\[6pt] & {} + (1-q^{-7}) (1-q^{-6}) (1-q^{-5}) (1-q^8) (1-q^9) (1-q^{10})(1-x) (1-qx) (1-xq^2) q^3 (1-q)^{-2} (1-q^2)^{-2} (1-q^3)^{-2} \\[6pt] & {} + (1-q^{-7}) (1-q^{-6}) (1-q^{-5}) (1-q^{-4}) (1-q^8) (1-q^9) (1-q^{10}) (1-q^{11}) (1-x) (1-qx) \left( 1-x{q}^{2} \right) \left( 1-x{q}^{3} \right) {q}^{4} \left( 1 -q \right) ^{-2} \left( 1-{q}^{2} \right) ^{-2} \left( 1-{q}^{3} \right) ^{-2} \left( 1-{q}^{4} \right) ^{-2}+ \left( 1-{q}^{-7} \right) \left( 1-{q}^{-6} \right) \left( 1-{q}^{-5} \right) \left( 1-{q}^{-4} \right) \left( 1-{q}^{-3} \right) \left( 1-{q}^{8 } \right) \left( 1-{q}^{9} \right) \left( 1-{q}^{10} \right) \left( 1-{q}^{11} \right) \left( 1-{q}^{12} \right) \left( 1-x \right) \left( 1-qx \right) \left( 1-x{q}^{2} \right) \left( 1-x{q }^{3} \right) \left( 1-x{q}^{4} \right) {q}^{5} \left( 1-q \right) ^{ -2} \left( 1-{q}^{2} \right) ^{-2} \left( 1-{q}^{3} \right) ^{-2} \left( 1-{q}^{4} \right) ^{-2} \left( 1-{q}^{5} \right) ^{-2}+ \left( 1-{q}^{-7} \right) \left( 1-{q}^{-6} \right) \left( 1-{q}^{- 5} \right) \left( 1-{q}^{-4} \right) \left( 1-{q}^{-3} \right) \left( 1-{q}^{-2} \right) \left( 1-{q}^{8} \right) \left( 1-{q}^{9} \right) \left( 1-{q}^{10} \right) \left( 1-{q}^{11} \right) \left( 1-{q}^{12} \right) \left( 1-{q}^{13} \right) \left( 1-x \right) \left( 1-qx \right) \left( 1-x{q}^{2} \right) \left( 1-x{q }^{3} \right) \left( 1-x{q}^{4} \right) \left( 1-x{q}^{5} \right) {q }^{6} \left( 1-q \right) ^{-2} \left( 1-{q}^{2} \right) ^{-2} \left( 1 -{q}^{3} \right) ^{-2} \left( 1-{q}^{4} \right) ^{-2} \left( 1-{q}^{5} \right) ^{-2} \left( 1-{q}^{6} \right) ^{-2}+ \left( 1-{q}^{-7} \right) \left( 1-{q}^{-6} \right) \left( 1-{q}^{-5} \right) \left( 1-{q}^{-4} \right) \left( 1-{q}^{-3} \right) (1-q^{-2}) (1-q^{-1}) (1-q^8) (1-q^9) (1-q^{10}) (1-q^{11}) (1-q^{12}) (1-q^{13}) (1-q^{14}) (1-x) (1-qx) (1-xq^2) (1-xq^3) (1-xq^4) (1-xq^5) (1-xq^6) q^7 (1-q)^{-2} (1-q^2)^{-2} (1-q^3)^{-2} (1-q^4)^{-2} (1-q^5)^{-2} ( 1-q^6 )^{-2} (1-q^7)^{-2}\cdots \end{align} \)
then let q → 1:
\( qL2=\lim_{q \to 1}qL=-1+56x+3432x^7-12012x^6+16632x^5-11550x^4+ 4200x^3-756x^2 \)
On the other hand
\( P_7(2x-1)=-1+56x+3432x^7-12012x^6+16632x^5-11550x^4+ 4200x^3-756x^2 \)
Obviously \( qL2 = P_7(2x-1) \) QED.
References
Roelof Koekoek, Peter Lesky, Rene Swattouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues, p 443,Springer
Andrews, George E.; Askey, Richard (1985), "Classical orthogonal polynomials", in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A., Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag, pp. 36–62, doi:10.1007/BFb0076530, ISBN 978-3-540-16059-5, MR 838970
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
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