Recurrence relations, continued fractions and orthogonal polynomials.

*(English)*Zbl 0548.33001
Mem. Am. Math. Soc. 300, 108 p. (1984).

The introduction and chapter 1 present an overview of the subject along somewhat classical lines, but the treatment identifies the sympathies of the authors with the zeros of a distinctly contemporary component of the theory of orthogonal polynomials, mainly, the Pollaczek polynomials. These polynomials, which have obsessed many mathematicians, present a number of pathological features. They are not hypergeometric in type (except in the most liberal sense of the phrase), and their weight function vanishes so strongly at the endpoint that it fails to satisfy Szegö’s integrability condition.

Those special functions which are needed for subsequent developments are introduced here, a good feature. A discussion of the properties of the spectrum of the distribution function is particularly worthwhile. So much more is now known about this topic than it was when Szegö wrote his book, and the discussion in the Bateman manuscript volumes is too cursory to be of much use.

Chapter 3 introduces the Al-Salam-Chihara polynomials. Al-Salam and Chihara, seeking to generalize certain properties of the Laguerre polynomials asked for what pairs of orthogonal polynomials \(\{p_ n(x)\}\), \(\{q_ n(x)\}\) the function \(Q_ n\) defined by \(Q_ n(x,y)=\sum^{n}_{k=0}p_ k(x)q_{n-k}(y)\) is also a sequence of orthogonal polynomials in x for infinitely many values of y - a highly ingenious way to extrapolate a mathematical set; in special functions it is all in knowing the right question to ask. The two workers were able to characterize the polynomials and compute their distribution function. The investigation led to polynomials which satisfy \[ P_{n+1}(x)=(x-aq^ n)P_ n(x)-(c-bq^{n-1})(1-q^ n)P_{n-1}(x), \] and the investigators were able to obtain their distribution function. The present authors reproduce that derivation, which requires obtaining asymptotic formulas for the polynomials for large n, and give some explicit formulas for the polynomials which involve the basic function \({}_ 2\phi_ 1\). They then establish the connection with Meixner- Pollaczek polynomials.

In chapter 4 the author discuss a fascinating class of polynomials whose special cases arise in certain birth and death processes, the so-called random walk polynomials. Special cases of these polynomials have been investigated by a number of writers: Karlin, MacGregor, Carlitz, Pollaczek, Szegö.

Chapter 5 has a very welcome discussion of the Hadamard integral. This relatively little-known analytic tool, which provides a way to generalize integrals with integrands having logarithmic singularities (somewhat in the same way that the gamma function generalizes integrals of functions having algebraic singularities) was the crucial weapon that allowed Pollaczek to obtain the generating function for the polynomials that bear his name. The authors subsequently use this tool to compute the singular part of the generating function for the polynomials they study, then apply Darboux’s method to obtain asymptotic formulas, which ultimately enables them to derive the distribution function.

Chapters 6 and 7 treat the random walk polynomials and certain of their q-extensions. Various very interesting cases arise wherein the distribution function may have point masses. Those who like the subject will follow the intriguing process of deriving the distribution function.

The presentation of the material and the manner of development displays all the characteristics we have come to associate with the writing of these two mathematicians. Few mathematicians have such a cultured approach to their field, one so securely ancored in a preception of historical trends and historical facts. The writing is smooth and uncluttered. Outstanding problems are mentioned wherever they occur (note Ph.D. candidates looking for thesis material!).

Chapter 9, in fact, itemizes a number of open problems associated with the polynomials the authors have studied. The authors state, ”we do not have enough examples of the different types of weight functions that can occur under different assumptions on the coefficients in the recurrence relation (for the polynomials)”.

Indeed this is the problem in the theory of orthogonal polynomials. The lack of sufficiently comprehensive conditions on the coefficients that will preclude the appearance of point masses in the distribution function have stymied a number of investigations (including some of the reviewer’s). The reader who is new to q polynomials will find this an exciting introduction to the subject, the first to appear in book form. Further, the formulas the author use are elegant, simple, and wisely chosen. Neither author has ever espoused generalization for the sake of generalization. One continually has the sense that the problems explored are closely related to matters of real concern in mathematics and the physical sciences.

Those special functions which are needed for subsequent developments are introduced here, a good feature. A discussion of the properties of the spectrum of the distribution function is particularly worthwhile. So much more is now known about this topic than it was when Szegö wrote his book, and the discussion in the Bateman manuscript volumes is too cursory to be of much use.

Chapter 3 introduces the Al-Salam-Chihara polynomials. Al-Salam and Chihara, seeking to generalize certain properties of the Laguerre polynomials asked for what pairs of orthogonal polynomials \(\{p_ n(x)\}\), \(\{q_ n(x)\}\) the function \(Q_ n\) defined by \(Q_ n(x,y)=\sum^{n}_{k=0}p_ k(x)q_{n-k}(y)\) is also a sequence of orthogonal polynomials in x for infinitely many values of y - a highly ingenious way to extrapolate a mathematical set; in special functions it is all in knowing the right question to ask. The two workers were able to characterize the polynomials and compute their distribution function. The investigation led to polynomials which satisfy \[ P_{n+1}(x)=(x-aq^ n)P_ n(x)-(c-bq^{n-1})(1-q^ n)P_{n-1}(x), \] and the investigators were able to obtain their distribution function. The present authors reproduce that derivation, which requires obtaining asymptotic formulas for the polynomials for large n, and give some explicit formulas for the polynomials which involve the basic function \({}_ 2\phi_ 1\). They then establish the connection with Meixner- Pollaczek polynomials.

In chapter 4 the author discuss a fascinating class of polynomials whose special cases arise in certain birth and death processes, the so-called random walk polynomials. Special cases of these polynomials have been investigated by a number of writers: Karlin, MacGregor, Carlitz, Pollaczek, Szegö.

Chapter 5 has a very welcome discussion of the Hadamard integral. This relatively little-known analytic tool, which provides a way to generalize integrals with integrands having logarithmic singularities (somewhat in the same way that the gamma function generalizes integrals of functions having algebraic singularities) was the crucial weapon that allowed Pollaczek to obtain the generating function for the polynomials that bear his name. The authors subsequently use this tool to compute the singular part of the generating function for the polynomials they study, then apply Darboux’s method to obtain asymptotic formulas, which ultimately enables them to derive the distribution function.

Chapters 6 and 7 treat the random walk polynomials and certain of their q-extensions. Various very interesting cases arise wherein the distribution function may have point masses. Those who like the subject will follow the intriguing process of deriving the distribution function.

The presentation of the material and the manner of development displays all the characteristics we have come to associate with the writing of these two mathematicians. Few mathematicians have such a cultured approach to their field, one so securely ancored in a preception of historical trends and historical facts. The writing is smooth and uncluttered. Outstanding problems are mentioned wherever they occur (note Ph.D. candidates looking for thesis material!).

Chapter 9, in fact, itemizes a number of open problems associated with the polynomials the authors have studied. The authors state, ”we do not have enough examples of the different types of weight functions that can occur under different assumptions on the coefficients in the recurrence relation (for the polynomials)”.

Indeed this is the problem in the theory of orthogonal polynomials. The lack of sufficiently comprehensive conditions on the coefficients that will preclude the appearance of point masses in the distribution function have stymied a number of investigations (including some of the reviewer’s). The reader who is new to q polynomials will find this an exciting introduction to the subject, the first to appear in book form. Further, the formulas the author use are elegant, simple, and wisely chosen. Neither author has ever espoused generalization for the sake of generalization. One continually has the sense that the problems explored are closely related to matters of real concern in mathematics and the physical sciences.

Reviewer: J.Wimp

##### MSC:

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

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