In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. They are named for Leopold Gegenbauer (1849-1903). They are obtained from hypergeometric series in cases where the series is in fact finite: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

where is the falling factorial. (See Abramowitz & Stegun p561) In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

References

• Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York:Dover. ISBN 0486612724. (See chapter 22)