## Generating functions of Legendre polynomials: A tribute to Fred Brafman.(English)Zbl 1242.33018

The authors present a generalization of Bailey’s identity for Appell’s hypergeometric function and its implication to generating functions of Legendre polynomials of the form $$\sum_{n=0}^{\infty}u_{n}P_{n}(x)z^{n}$$, where $$u_{n}$$ is an Apéry-like sequence, that is, a sequence satisfying $$(n+1)^2u_{n+1}=(an^2+an+b)u_n-cn^2u_{n-1}$$, where $$n\geq 0$$ and $$u_{-1}=0, u_0=1$$. The authors also give generating functions for rarefied Legendre polynomials and construct a new family of identities for $$1/\pi$$.

### MSC:

 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33C20 Generalized hypergeometric series, $${}_pF_q$$
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### References:

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