Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-3-642-05013-8/hbk; 978-3-642-26351-4/pbk; 978-3-642-05014-5/ebook). xix, 578 p. EUR 99.95/net; £ 90.00; SFR 155.50 (2010).
Let be a linear differential or a difference operator of second order with polynomial coefficients. The problem is to find conditions on the coefficients of to guarantee the following. For any integer there exist a number and a polynomial
such that . This problem is considered in the chapters 4–6 of the book.
The analogous problem with
is considered in chapters 7–8 of the book.
Let , , , . The hypergeometric function is defined by the series
The polynomials of Wilson and the polynomials of Racah are defined by the formulas
, where and or or
Limits of the type are considered. For example,
where are Jacobi polynomials.
It is an interesting case when is a polynomial of degree depending on a set of parameters as in the example. In this case one can repeat the limiting process. Let be a set of polynomials obtained in such a way starting with and . Let us take sheet of paper and write on it symbols of polynomials from . We join some of the symbols by arrows. The figure means that the polynomial is obtained from by the limiting process. The poster we obtain is called the Askey scheme. We see the Askey scheme on the page 189 of the book. This scheme is grounded in the chapter 9.
-analogues of differential and difference operators, -analogues of transcendental functions are known in mathematics. There are -analogues of the described results in the second part (chapters 10–14) of the book.
It is pertinent to quote Askey: “A set of orthogonal polynomials is classical if it is a special case or a limiting case of the Askey-Wilson polynomials or -Racah polynomials.”