On Gegenbauer polynomials and coefficients \(c^{\ell}_{j}(\nu )(1\leq j\leq \ell, \nu >-1/2)\). (English) Zbl 1376.33003
Summary: The Gegenbauer coefficients \(c^{\ell}_{j}(\nu )\) (\(1\leq j\leq \ell\), \(\nu >-1/2\)) appear in the Maclaurin expansion of the heat kernels on the \(n\)-sphere and the real projective \(n\)-space. In this note we show that these coefficients can be computed by transforming the higher order derivative formula for the Gegenbauer polynomials \(C_{k}^{\nu }\) (\(k\geq 0\), \(\nu >-1/2\)) into a spectral sum involving the powers of the eigenvalues of the associated Gegenbauer operator. We present explicit computations and various implications.
MSC:
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 35K08 | Heat kernel |
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