##
**The umbral calculus.**
*(English)*
Zbl 0536.33001

A Sheffer sequence is a set of polynomials \(\{s_n(x)\}\) with the generating function
\[
\sum^{\infty}_{n=0}s_n(x)z^n/n!=A(z)\exp(xB(z)), \tag{1}
\]
where \(A(0)\neq 0\), \(B(0)=0\), \(B'(0)\neq 0\), and \(A(z)\) and \(B(z)\) are analytic in a neighborhood of the origin, or more generally have formal power series expansions about zero. Most of this book is a study of Sheffer sequences via an operational method known as umbral calculus. Following Rota, the author treats the umbral calculus via linear functionals and operators. If \(f(t)\) has the formal power series \(\sum a_n t^n/n!\) then \(\langle f(t)| x_n\rangle=a_n\). I would rather have seen the author use \(D\) as the variable since that is the usual notation for a derivative, but with a bit of effort one can get used to thinking of \(t\) as \(d/dx\). A linear operator associated with the formal power series of \(f\) is defined by \(f(t)x^n=\sum \binom nk a_k x^{n-k}\). In this setting Sheffer sequences can be defined as follows. Assume \(a_0=0\), \(a_1\neq 0\) and \(g(0)\neq 0\). Then there is a unique sequence of polynomials \(s_n(x)\) satisfying \(\langle g(t)[f(t)]^k| s_n(x)\rangle =n!\delta_{n,k}\), \(n,k\geq 0\). These polynomials are the Sheffer sequence for the pair \((g(t),f(t))\). The generating function (1) can be written as
\[
[g(z)]^{-1}e^{xz}=\sum^{\infty}_{n=0}s_n(x)[f(z)]^ n/n!\, .
\]

Many expansions and other facts are derived for Sheffer sequences, including special cases such as Appell sequences (those when \(f(t)=t)\). Among the examples given to illustrate the general theory are \(x^n\), \(x(x-1)\cdots(x-n+1)\), the polynomials discovered by Euler that are called Abel polynomials, those found by Lambert and called Gould polynomials, Hermite polynomials, Laguerre polynomials, Meixner polynomials, Charlier polynomials, Bessel polynomials, Bell polynomials, Bernoulli polynomials, Euler polynomials, and many miscellaneous polynomials mentioned earlier by Boas and Buck and in the Bateman project.

The last two chapters are an introduction to special topics including the Lagrange inversion formula, expansions of one Sheffer sequence in terms of another if they are sufficiently similar, a proof of Meixner’s theorem giving all sets of orthogonal polynomials which are Sheffer sequences, and a brief introduction to other umbral calculi, including a very brief introduction to \(q\)-umbral calculus. The most satisfying \(q\)-umbral calculus has recently been discovered by A. M. Garsia and J. Remmel [Houston J. Math. 12, 503–523 (1986; Zbl 0616.05006)]. Here is a test question for those who study \(q\)-umbral methods. The right \(q\)-extension of (1) is probably \[ \sum s_n(x)t^n=A(t)\prod^{\infty}_{0}[1-xB(tq^n)] \] with \(A(t)\) and \(B(t)\) as given above. It is likely that all orthogonal polynomials in this set were found by W. A. Al-Salam and T. S. Chihara [SIAM J. Math. Anal. 7, 16–28 (1976; Zbl 0323.33007)]. An adequate \(q\)-umbral calculus should be able to show this.

There are some misprints. A few noted are: \(\binom yk\) should be \(y^k/k!\) twice on page 55, a minus sign is missing in the first formula on page 61, \(e\nu^{t^ 2/2}\) should be \(e^{\nu t^2/2}\) on page 158. The infinite \(q\)-binomial theorem on page 179 is stated incorrectly twice. In the standard notation the only error is \(z\) instead of \(y\) on the left hand side. In the alternative notation used in the middle formula of this page, \(z\) should be \(x\) on the right hand side and the author forgot to replace \(t\) by \(t/(1-q)\) in the series, as he said to in the previous line. Also on page 180, the formula for \(H_n(x;y)\) in the middle of the page is at most a \(q\)-analogue of \((x+y)^n\), not the \(q\)-analog. A better \(q\)- analog was given on the previous page.

A couple of statements that can be misinterpreted were given. On page 78 it is said that \(y_n(x)\) and their derivatives form an orthogonal sequence of polynomials. What is true is that \(\{y_n(x)\}\) are orthogonal with respect to a signed measure. That is also true for \(\{y^{(k)}_{n+k}(x)\}\) for each fixed \(k\), and even more generally for \(\{{}_2F_1(-n,n+a;-;-x/2)\}.\) On page 159 the author says “From these examples the reader should have no difficulty solving other recurrence relations (whose solutions are not known in advance).” A novice might conclude from this that one can exactly solve most or all linear recurrence relations. That is far from the case, although there are quite a few interesting ones that are not given in the standard handbooks.

Much of the notation annoyed me. The use of \(t\) instead of \(D\) was a minor instance. Much more annoying was the use of the standard letters to denote standard functions with a different normalization. Thus \(L_n^{\alpha}(x)\) does not denote the usual Laguerre polynomials but \(n!\) times the usual ones. \(H_n(x)\) does not denote the Hermite polynomials as given by Szegő and Erdélyi in the two standard references for the classical orthogonal polynomials, but polynomials orthogonal with respect to \(\exp(-x^2/2).\) I know why the author changed these functions, but do not understand why he did not change the notation as well. \(\ell_n^{\alpha}(x)\) would have been better than \(L_n^{\alpha}(x)\). Hypergeometric functions occur in many of the examples, but the standard notation was not used.

The worst case of confusion occurs in the use of \((x)_ n\) to denote \(x(x-1)\cdots(x-n+1),\) as many combinatorialists do, rather than \(x(x+1)\cdots(x+n-1),\) as almost everyone else does. The standard \(q\)-notation \((a;q)_ n=(1-a)(1-aq)\cdots(1-aq^{n-1})\) is introduced and used in the statement of the \(q\)-binomial theorem (due to Rothe, not Heine), but the notation that was used in most of the \(q\)-formulas, \([x]_{y,n}\) for \(x^n(y/x;q)_n\) is an abomination and should not be complied by others.

The bibliography is probably relatively good when dealing with umbral calculus, but is lacking in references to other treatments of operational calculus and any of the deeper formulas for special functions related to additional formulas. Two examples are I. I. Hirschman and D. V. Widder’s very important and unjustly neglected book “The convolution transform.” Princeton: Princeton University Press (1955; Zbl 0065.09301) for some operational results and Ch. F. Dunkl’s work on Krawtchouk polynomials [Indiana Univ. Math. J. 25, 335–358 (1976; Zbl 0326.33008)]. Without some comments on other work a novice could get a wrong idea about how much can be done with the present methods. They work fine for a limited class of problems, and for some they seem to be the best method. But there are many very important sets of polynomials which can not be treated yet, and most of the more complicated formulas for some Sheffer polynomials do not seem to be amenable to umbral methods.

Many expansions and other facts are derived for Sheffer sequences, including special cases such as Appell sequences (those when \(f(t)=t)\). Among the examples given to illustrate the general theory are \(x^n\), \(x(x-1)\cdots(x-n+1)\), the polynomials discovered by Euler that are called Abel polynomials, those found by Lambert and called Gould polynomials, Hermite polynomials, Laguerre polynomials, Meixner polynomials, Charlier polynomials, Bessel polynomials, Bell polynomials, Bernoulli polynomials, Euler polynomials, and many miscellaneous polynomials mentioned earlier by Boas and Buck and in the Bateman project.

The last two chapters are an introduction to special topics including the Lagrange inversion formula, expansions of one Sheffer sequence in terms of another if they are sufficiently similar, a proof of Meixner’s theorem giving all sets of orthogonal polynomials which are Sheffer sequences, and a brief introduction to other umbral calculi, including a very brief introduction to \(q\)-umbral calculus. The most satisfying \(q\)-umbral calculus has recently been discovered by A. M. Garsia and J. Remmel [Houston J. Math. 12, 503–523 (1986; Zbl 0616.05006)]. Here is a test question for those who study \(q\)-umbral methods. The right \(q\)-extension of (1) is probably \[ \sum s_n(x)t^n=A(t)\prod^{\infty}_{0}[1-xB(tq^n)] \] with \(A(t)\) and \(B(t)\) as given above. It is likely that all orthogonal polynomials in this set were found by W. A. Al-Salam and T. S. Chihara [SIAM J. Math. Anal. 7, 16–28 (1976; Zbl 0323.33007)]. An adequate \(q\)-umbral calculus should be able to show this.

There are some misprints. A few noted are: \(\binom yk\) should be \(y^k/k!\) twice on page 55, a minus sign is missing in the first formula on page 61, \(e\nu^{t^ 2/2}\) should be \(e^{\nu t^2/2}\) on page 158. The infinite \(q\)-binomial theorem on page 179 is stated incorrectly twice. In the standard notation the only error is \(z\) instead of \(y\) on the left hand side. In the alternative notation used in the middle formula of this page, \(z\) should be \(x\) on the right hand side and the author forgot to replace \(t\) by \(t/(1-q)\) in the series, as he said to in the previous line. Also on page 180, the formula for \(H_n(x;y)\) in the middle of the page is at most a \(q\)-analogue of \((x+y)^n\), not the \(q\)-analog. A better \(q\)- analog was given on the previous page.

A couple of statements that can be misinterpreted were given. On page 78 it is said that \(y_n(x)\) and their derivatives form an orthogonal sequence of polynomials. What is true is that \(\{y_n(x)\}\) are orthogonal with respect to a signed measure. That is also true for \(\{y^{(k)}_{n+k}(x)\}\) for each fixed \(k\), and even more generally for \(\{{}_2F_1(-n,n+a;-;-x/2)\}.\) On page 159 the author says “From these examples the reader should have no difficulty solving other recurrence relations (whose solutions are not known in advance).” A novice might conclude from this that one can exactly solve most or all linear recurrence relations. That is far from the case, although there are quite a few interesting ones that are not given in the standard handbooks.

Much of the notation annoyed me. The use of \(t\) instead of \(D\) was a minor instance. Much more annoying was the use of the standard letters to denote standard functions with a different normalization. Thus \(L_n^{\alpha}(x)\) does not denote the usual Laguerre polynomials but \(n!\) times the usual ones. \(H_n(x)\) does not denote the Hermite polynomials as given by Szegő and Erdélyi in the two standard references for the classical orthogonal polynomials, but polynomials orthogonal with respect to \(\exp(-x^2/2).\) I know why the author changed these functions, but do not understand why he did not change the notation as well. \(\ell_n^{\alpha}(x)\) would have been better than \(L_n^{\alpha}(x)\). Hypergeometric functions occur in many of the examples, but the standard notation was not used.

The worst case of confusion occurs in the use of \((x)_ n\) to denote \(x(x-1)\cdots(x-n+1),\) as many combinatorialists do, rather than \(x(x+1)\cdots(x+n-1),\) as almost everyone else does. The standard \(q\)-notation \((a;q)_ n=(1-a)(1-aq)\cdots(1-aq^{n-1})\) is introduced and used in the statement of the \(q\)-binomial theorem (due to Rothe, not Heine), but the notation that was used in most of the \(q\)-formulas, \([x]_{y,n}\) for \(x^n(y/x;q)_n\) is an abomination and should not be complied by others.

The bibliography is probably relatively good when dealing with umbral calculus, but is lacking in references to other treatments of operational calculus and any of the deeper formulas for special functions related to additional formulas. Two examples are I. I. Hirschman and D. V. Widder’s very important and unjustly neglected book “The convolution transform.” Princeton: Princeton University Press (1955; Zbl 0065.09301) for some operational results and Ch. F. Dunkl’s work on Krawtchouk polynomials [Indiana Univ. Math. J. 25, 335–358 (1976; Zbl 0326.33008)]. Without some comments on other work a novice could get a wrong idea about how much can be done with the present methods. They work fine for a limited class of problems, and for some they seem to be the best method. But there are many very important sets of polynomials which can not be treated yet, and most of the more complicated formulas for some Sheffer polynomials do not seem to be amenable to umbral methods.

Reviewer: Richard Askey (Madison)

### MSC:

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

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

05A40 | Umbral calculus |

05A10 | Factorials, binomial coefficients, combinatorial functions |

05A19 | Combinatorial identities, bijective combinatorics |

05A15 | Exact enumeration problems, generating functions |

05A30 | \(q\)-calculus and related topics |

44A55 | Discrete operational calculus |