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We have already seen in the chapter on Jacobi polynomials (Ch. 8) that many special functions such as Hermite, Laguerre, Chebyshev, etc., are special cases of Jacobi polynomials. These special functions are solutions to specific second-order differential equations. Is there a general second-order ordinary differential equation that reduces to these specific differential equations, in other words, the grand-daddy of them all? The answer to this is yes, the so-called hypergeometric equation, also known as Gauss’s differential equation (named after Carl Fredrich Gauss, 1777–1855, “The prince of mathematicians”) and the confluent hypergeometric equation. If we make suitable transformations of the independent variable, the solutions can represent the special functions with additional multiplying factors.
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