##
**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. (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

\[ _rF_s \left( \begin{matrix} a_1,\ldots, a_r \\ b_1,\ldots, b_s \\ \end{matrix}; z\right) =\sum_{k=0}^\infty \frac{(a_1, \dots , a_r)_k}{(b_1, \dots , b_s)_k}\frac{z^k}{k!}. \]

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

\[ \begin{aligned} \frac{W_n(x^2, a, b, c, d)}{(a+b)_n(a+c)_n(a+d)_n} &= {}_4F_3 \left( \begin{matrix} -n, & n+a+b+c+d -1 , & a+ix,\;a-ix \\ a+b, & a+c , & a+d \\ \end{matrix}; 1 \right), \\ R_n(\lambda(x), \alpha , \beta , \gamma , \delta) &= {}_4F_3 \left( \begin{matrix} -n, & n+\alpha +\beta +1 , & -x,\;x+\gamma +\delta +1 \\ \alpha +1, & \beta + \delta +1 , & \gamma +1 \\ \end{matrix} ; 1 \right), \end{aligned} \]

\(n=0, 1, \dots , N\), 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.”

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

\[ _rF_s \left( \begin{matrix} a_1,\ldots, a_r \\ b_1,\ldots, b_s \\ \end{matrix}; z\right) =\sum_{k=0}^\infty \frac{(a_1, \dots , a_r)_k}{(b_1, \dots , b_s)_k}\frac{z^k}{k!}. \]

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

\[ \begin{aligned} \frac{W_n(x^2, a, b, c, d)}{(a+b)_n(a+c)_n(a+d)_n} &= {}_4F_3 \left( \begin{matrix} -n, & n+a+b+c+d -1 , & a+ix,\;a-ix \\ a+b, & a+c , & a+d \\ \end{matrix}; 1 \right), \\ R_n(\lambda(x), \alpha , \beta , \gamma , \delta) &= {}_4F_3 \left( \begin{matrix} -n, & n+\alpha +\beta +1 , & -x,\;x+\gamma +\delta +1 \\ \alpha +1, & \beta + \delta +1 , & \gamma +1 \\ \end{matrix} ; 1 \right), \end{aligned} \]

\(n=0, 1, \dots , N\), 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.”

Reviewer: Anatoly Filip Grishin (Khar’kov)

### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |

### Keywords:

classical orthogonal polynomials; Hahn polynomials; Askey-Wilson polynomials; Racah polynomials; Askey scheme
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\textit{R. Koekoek} et al., Hypergeometric orthogonal polynomials and their \(q\)-analogues. With a foreword by Tom H. Koornwinder. Berlin: Springer (2010; Zbl 1200.33012)

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### Digital Library of Mathematical Functions:

(17.4.2) ‣ §17.4(i) {_𝑟}ϕ_𝑠 Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 𝑞-Hypergeometric and Related Functions§18.1(iii) Other Notations ‣ §18.1 Notation ‣ Notation ‣ Chapter 18 Orthogonal Polynomials

§18.22(ii) Difference Equations in 𝑥 ‣ §18.22 Hahn Class: Recurrence Relations and Differences ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials

§18.26(iii) Difference Relations ‣ §18.26 Wilson Class: Continued ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials

§18.27(iii) Big 𝑞-Jacobi Polynomials ‣ §18.27 𝑞-Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.27(ii) 𝑞-Hahn Polynomials ‣ §18.27 𝑞-Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.27(i) Introduction ‣ §18.27 𝑞-Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.27(iv) Little 𝑞-Jacobi Polynomials ‣ §18.27 𝑞-Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.27(v) 𝑞-Laguerre Polynomials ‣ §18.27 𝑞-Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.28(iii) Al-Salam–Chihara Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.28(i) Introduction ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

§18.28(viii) 𝑞-Racah Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

Profile René F. Swarttouw ‣ About the Project

Profile Roelof Koekoek ‣ About the Project