Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.

*(English)*Zbl 0572.33012
Mem. Am. Math. Soc. 319, 55 p. (1985).

This memoir is a detailed exposition of a new family of orthogonal polynomials which has five free parameters and a continuous weight distribution. Indeed, there are cases where this distribution has a finite discrete part in addition. The orthogonality relation is based on a new contour integral. To state the most important results some notation is needed: (throughout \(q\in {\mathbb{C}}\) and \(| q| <1)\)
\[
(a;q)_ n:=\prod^{n-1}_{j=0}(1-aq^ j),\quad n=0,1,2,...;e_ q(a):=\prod^{\infty}_{j=0}(1-aq^ j),
\]

\[ p_ n(x;a,b,c,d| q):=a^{-n}(ab;q)_ n(ac;q)_ n(ad;q)_ n 4\phi_ 3\left[ \begin{matrix} q^{-n},q^{n-1}abcd,ae^{i\theta},ae^{-i\theta};\\ ab,ac,ad\quad q,q\end{matrix} \right] \] where \(x=\cos \theta\), \(n=0,1,2,..\). (also denoted by \(p_ n(x))\), a terminating basic hypergeometric series.

The underlying integral depends on five parameters q, \(a_ j\), (1\(\leq j\leq 4)\) such that \(| q| <1\) and \(a_ ja_ k\neq q^{\ell}\) for \(\ell =0,1,2,...,(1\leq j,k\leq 4):\) \[ (1/2\pi i)\int_{C}e_ q(z^ 2)e_ q(z^{-2})\prod^{4}_{j=1}(e_ q(a_ jz)e_ q(a_ j/z))^{-1}(dz/z) \]

\[ =\frac{2e_ q(abcd)}{e_ q(q)}\prod_{1\leq j<k\leq 4}e_ q(a_ ja_ k)^{-1}, \] where C is a closed positively oriented contour consisting of the unit circle deformed so as to separate the sequences of poles converging to zero from the sequences of poles converging to infinity.

It is shown that \(\{p_ n(x)\}\) is a family of polynomials in x, with degree \((p_ n)=n\), and satisfying a three-term recurrence. Further, each \(p_ n\) is symmetric in a,b,c,d. The purely continuous weight distribution occurs for \(-1<q<1\), \(\max (| a|,| b|,| c|,| d|)<1\), and a,b,c,d all real or appearing in conjugate pairs: then \[ \int^{1}_{-1}p_ n(x)p_ m(x)w(x)(1-x^ 2)^{- 1/2}dx=0\quad for\quad n\neq m, \] where \[ w(x)=e_ q(e^{2i\theta})e_ q(e^{-2i\theta})\prod^{4}_{j=1}(e_ q(a_ je^{i\theta})e_ q(a_ je^{-i\theta}))^{-1},\quad where\quad x=\cos \theta, \] and \(a_ j\) (1\(\leq j\leq 4)\) takes the values a,b,c,d. The integral \[ \int^{1}_{-1}p_ n(x)^ 2w(x)(1-x^ 2)^{-1/2}dx \] is explicitly found. When any of the parameters a,b,c,d exceed 1, a finite discrete part (explicitly known) is added to the weight distribution.

The power of these results stems from the large number of free parameters. Special choices lead to previously studied families. For example, the case \(c=-a\), \(b=aq^{1/2}=-d\) gives the Rogers continuous q-ultraspherical polynomials (R. Askey and M. E.-H. Ismail, Studies in pure mathematics, Mem. of P. Turán, 55-78 (1983; Zbl 0532.33006).

Another example comes from the choice \(a=q^{\alpha /2+1/4}\), \(c=q^{1/2}\), \(b=-q^{\beta /2+1/4}\), \(d=bq^{1/2}\); this is the family of little q-Jacobi polynomials studied by G. Andrews and R. Askey.

Other special cases are also discussed in the paper, including the example \(q=0\), and a q-analogue of Meixner-Pollaczek polynomials. There is a Rodrigues type formula for \(p_ n\) which depends on a divided difference operator. Also, the connection coefficients between two different families \(\{p_ n(x;a,b,c,d| q)\}\) and \(\{p_ n(x;a',b',c',d'| q)\}\) are given as \({}_ 5\phi_ 4\)-series, and some tractable special cases are discussed. This is a paper of fundamental importance in the theory of orthogonal polynomials in one variable of hypergeometric type.

\[ p_ n(x;a,b,c,d| q):=a^{-n}(ab;q)_ n(ac;q)_ n(ad;q)_ n 4\phi_ 3\left[ \begin{matrix} q^{-n},q^{n-1}abcd,ae^{i\theta},ae^{-i\theta};\\ ab,ac,ad\quad q,q\end{matrix} \right] \] where \(x=\cos \theta\), \(n=0,1,2,..\). (also denoted by \(p_ n(x))\), a terminating basic hypergeometric series.

The underlying integral depends on five parameters q, \(a_ j\), (1\(\leq j\leq 4)\) such that \(| q| <1\) and \(a_ ja_ k\neq q^{\ell}\) for \(\ell =0,1,2,...,(1\leq j,k\leq 4):\) \[ (1/2\pi i)\int_{C}e_ q(z^ 2)e_ q(z^{-2})\prod^{4}_{j=1}(e_ q(a_ jz)e_ q(a_ j/z))^{-1}(dz/z) \]

\[ =\frac{2e_ q(abcd)}{e_ q(q)}\prod_{1\leq j<k\leq 4}e_ q(a_ ja_ k)^{-1}, \] where C is a closed positively oriented contour consisting of the unit circle deformed so as to separate the sequences of poles converging to zero from the sequences of poles converging to infinity.

It is shown that \(\{p_ n(x)\}\) is a family of polynomials in x, with degree \((p_ n)=n\), and satisfying a three-term recurrence. Further, each \(p_ n\) is symmetric in a,b,c,d. The purely continuous weight distribution occurs for \(-1<q<1\), \(\max (| a|,| b|,| c|,| d|)<1\), and a,b,c,d all real or appearing in conjugate pairs: then \[ \int^{1}_{-1}p_ n(x)p_ m(x)w(x)(1-x^ 2)^{- 1/2}dx=0\quad for\quad n\neq m, \] where \[ w(x)=e_ q(e^{2i\theta})e_ q(e^{-2i\theta})\prod^{4}_{j=1}(e_ q(a_ je^{i\theta})e_ q(a_ je^{-i\theta}))^{-1},\quad where\quad x=\cos \theta, \] and \(a_ j\) (1\(\leq j\leq 4)\) takes the values a,b,c,d. The integral \[ \int^{1}_{-1}p_ n(x)^ 2w(x)(1-x^ 2)^{-1/2}dx \] is explicitly found. When any of the parameters a,b,c,d exceed 1, a finite discrete part (explicitly known) is added to the weight distribution.

The power of these results stems from the large number of free parameters. Special choices lead to previously studied families. For example, the case \(c=-a\), \(b=aq^{1/2}=-d\) gives the Rogers continuous q-ultraspherical polynomials (R. Askey and M. E.-H. Ismail, Studies in pure mathematics, Mem. of P. Turán, 55-78 (1983; Zbl 0532.33006).

Another example comes from the choice \(a=q^{\alpha /2+1/4}\), \(c=q^{1/2}\), \(b=-q^{\beta /2+1/4}\), \(d=bq^{1/2}\); this is the family of little q-Jacobi polynomials studied by G. Andrews and R. Askey.

Other special cases are also discussed in the paper, including the example \(q=0\), and a q-analogue of Meixner-Pollaczek polynomials. There is a Rodrigues type formula for \(p_ n\) which depends on a divided difference operator. Also, the connection coefficients between two different families \(\{p_ n(x;a,b,c,d| q)\}\) and \(\{p_ n(x;a',b',c',d'| q)\}\) are given as \({}_ 5\phi_ 4\)-series, and some tractable special cases are discussed. This is a paper of fundamental importance in the theory of orthogonal polynomials in one variable of hypergeometric type.

Reviewer: Ch.Dunkl

##### MSC:

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

33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |

39A10 | Additive difference equations |