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Special functions and the Bieberbach conjectures. (English) Zbl 0655.33002
When dealing with the uses of special functions, many users are content to quote formulas from handbooks. However there are still many people who like to understand how the formulas in the handbooks were derived. In the proof of an inequality about a generalized hypergeometric series, Gasper and I used Clausen’s formula for the square of a particular ${}_{2}{F}_{2}$ as a ${}_{3}{F}_{2}$, and Gegenbauer’s connection coefficient formula for two ultraspherical polynomials. Proofs of these two results are given. Clausen’s proof is given for his result, using the differential equation satisfied by ${y}^{2}$, ${z}^{2}$ and yz when y and z are solutions of a linear second order homogeneous differential equation. Two proofs of Gegenbauer’s formula are given. Both start with the special case when one parameter is infinite, which is obtained by differentiating the classical generating function to set up a recurrence relation. In fact, Gegenbauer’s formula follows directly from the generating function by differentiating K times.
 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 33C45 Orthogonal polynomials and functions of hypergeometric type