The Pearson Equation and the Beta Integrals

An alternate proof of the classical beta integral is given by using the Pearson equation whose solution converts the hypergeometric equation into a self-adjoint equation. A generalization of this idea to difference equations on linear lattices enables a proof of Barnes’s second lemma by using his first lemma and gives extensions of some of Ramanujan’s formulas. Similar analysis on q-quadratic lattices gives an extension of Askey’s integral on the real line and the corresponding basic bilateral sum of Gosper. Another q-quadratic lattice gives an extension of the Askey–Wilson integral. A quadratic lattice is used to evaluate a principal value integral on the real line, whose q-analogue is shown to be equivalent to the Askey integral extension. A list of various beta integrals is also presented.