zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hypergeometric orthogonal polynomials and their $q$-analogues. With a foreword by Tom H. Koornwinder. (English) Zbl 1200.33012
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 $L$ be a linear differential or a difference operator of second order with polynomial coefficients. The problem is to find conditions on the coefficients of $L$ to guarantee the following. For any integer $n\geq 0$ there exist a number $\lambda_n$ and a polynomial $$P_n(x)=x^n +c_{n-1} x^{n-1} + \dots +c_0$$ such that $ L P_n = \lambda_n P_n $. This problem is considered in the chapters 4--6 of the book. The analogous problem with $$P_n(x)=(x(x+u))^n +c_{n-1} (x(x+u))^{n-1} + \dots +c_0$$ is considered in chapters 7--8 of the book. Let $(a)_0=1 $, $ (a)_k= a(a+1)\cdot \dots \cdot (a+k-1)$, $k\geq 1$, $(a_1, \dots , a_r)_k = (a_1)_k \cdot \dots \cdot (a_r)_k$. The hypergeometric function $_rF_s$ is defined by the series $$_rF_s \left( \matrix a_1,\ldots, a_r \\ b_1,\ldots, b_s \\ \endmatrix; z\right) =\sum_{k=0}^\infty \frac{(a_1, \dots , a_r)_k}{(b_1, \dots , b_s)_k}\frac{z^k}{k!}.$$ The polynomials of Wilson $W_n(x, a, b, c, d)$ and the polynomials of Racah $R_n(x, \alpha , \beta , \gamma , \delta)$ are defined by the formulas $$\align \frac{W_n(x^2, a, b, c, d)}{(a+b)_n(a+c)_n(a+d)_n} &= {}_4F_3 \left( \matrix -n, & n+a+b+c+d -1 , & a+ix,\ a-ix \\ a+b, & a+c , & a+d \\ \endmatrix; 1 \right), \\ R_n(\lambda(x), \alpha , \beta , \gamma , \delta) &= {}_4F_3 \left( \matrix -n, & n+\alpha +\beta +1 , & -x,\ x+\gamma +\delta +1 \\ \alpha +1, & \beta + \delta +1 , & \gamma +1 \\ \endmatrix ; 1 \right), \endalign$$ $n=0, 1, \dots , N$, where $\lambda (x) = x(x+\gamma + \delta +1)$ and $\alpha +1 = -N$ or $\beta + \delta +1 = -N$ or $\gamma +1 = -N .$ Limits of the type $ Q_n=\lim_{t \to \infty} h_n(t)W_n \left(x(t), a(t), b(t), c(t), d(t)\right)$ are considered. For example, $$ \lim_{t\to\infty } \frac{W_n\left( \frac{1}{2}(1-x)t^2; \frac{1}{2}(\alpha +1); \frac{1}{2}(\alpha +1); \frac{1}{2}(\beta+1)+it;\frac{1}{2}(\beta+1)- it \right)}{t^{2n}n!} = P_n^{(\alpha, \beta)}(x),$$ where $P_n^{(\alpha, \beta)}(x)$ are Jacobi polynomials. It is an interesting case when $Q_n$ is a polynomial of degree $n$ depending on a set of parameters as in the example. In this case one can repeat the limiting process. Let $E$ be a set of polynomials obtained in such a way starting with $W_n(x, a, b, c, d)$ and $R_n(x, \alpha , \beta , \gamma , \delta)$. Let us take sheet of paper and write on it symbols of polynomials from $E$. We join some of the symbols by arrows. The figure $A \rightarrow B$ means that the polynomial $B$ is obtained from $A$ 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. $q$-analogues of differential and difference operators, $q$-analogues of transcendental functions are known in mathematics. There are $q$-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 $q$-Racah polynomials.”

33C45Orthogonal polynomials and functions of hypergeometric type
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Full Text: DOI