## The Pearson equation and the beta integrals.(English)Zbl 0808.33002

The Pearson equation is the first order linear, homogeneous differential equation with polynomial coefficients, the derivative multiplied by a second degree polynomial, and the other polynomial linear. K. Pearson was interested in this as a way of doing the various beta integrals and their limits all at once.
There are other beta type integrals, of Mellin-Barnes types, of Ramanujan type, sum analogues, and many $$q$$-extensions. Many of the known one dimensional beta and $$q$$-beta integrals and sums are treated by a common method in this paper. Some new ones are also obtained.