## Generalized $$q$$-Legendre polynomials.(English)Zbl 0797.33013

The author finds the polynomials $$u_ n$$ satisfying the 3-term recursion: $(1-q^{n+1}) (1+q^ n) u_{n+1} - f_ nu_ n + q^{2n- 1} (1-q^ n) (1+q^{N+1}) u_{n-1} = 0,$ where $f_ n = (1- q^{2n+1}) \left( 2q^ n-(1+q^ n) (1+q^{n+1}) \sum_{j=0}^ nq^{-jn} \left[ {n \over j} \right]_ q \left[ {n+j \over j} \right]_ qx_ j \right).$ For $$x_ 0=x$$, $$x_ j=0$$, $$j \geq 1$$, these give a $$q$$-analog of the Legendre polynomials: the $$\alpha = \beta = 1$$ case of the little $$q$$-Jacobi polynomials.

### MSC:

 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

### Keywords:

$$q$$-Legendre polynomials
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### References:

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