Gauss€™s hypergeometric ODE is

Here, a, b, c are constants. This ODE is of the form P_{2}y" + P_{1}y' + P_{0}y = 0, where P_{2}, P_{1}, P_{0 }are polynomials of degree 2, 1, 0, respectively. These polynomials are written so that the series solution takes a most practical form, namely,

This series is called the hyper geometric series. Its sum y_{1}(x) is called the hyper geometric function and is denoted by F(a, b, c; x). Here, c ‰ 0, -1, -2, . . . By choosing specific values of a, b, c we can obtain an incredibly large number of special functions as solutions of (15). This accounts for the importance of (15).

(a) Convergence. For what a or b will (16) reduce to a polynomial? Show that for any other a, b, c (c ‰ 0, -1, -2,. . .) the series (16) converges when |x| < 1.

(b) Show that

Find more such relations from the literature on special functions.

(c) Show that for r_{2} = 1 - c the Frobenius method yields the following solution (where c ‰ 2, 3, 4, . . . ):

Show that

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