In 1939, I. M. Sheffer published results regarding the characterizations of polynomials via general degree-lowering operators and showed that every polynomial sequence can be classified as belonging to exactly one Type. A large portion of his work was dedicated to developing results regarding the most basic type set, entitled Type 0.
In Chapter 1, a development of Sheffer’s characterization of the Type 0 orthogonal polynomial sequences is presented. First, Sheffer’s proof is addressed that every Type 0 set \(\{ P_n(x) \}_{n=0}^{\infty}\) can be characterized by the generating function \[ A(t) \exp{(x H(t))} = \sum_{n=0}^{\infty} P_n(x) t^n, \] where \(A(t)\) and \(H(t)\) are formal power series in \(t\) (they may or may not converge) which satisfy certain restrictions. Then, the additional theory is developed that Sheffer used in order to determine which Type 0 polynomial sequences are also orthogonal (continuous or discrete). These are now known to be the very well -studied and applicable Laguerre, Hermite, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials. These orthogonal polynomials are often called the Sheffer sequences. Then the three kinds of characterizations that Sheffer developed, which are entitled A-Type, B-Type and C-Type, are discussed. The definition of Type depends on which characterization of Type 0 has to be generalized. That is, each of the Sheffer Types generalizes a certain Type 0 characterizing structure. Moreover, a summary of the \(\sigma\)-Type classification developed by E. D. Rainville, which is a natural extension of the Sheffer A-Type classification, is presented. Since, in 1934, J. Meixner initially studied the generating function characterizing the Type 0 polynomial sequences and also determined which sets were orthogonal using a different approach than Sheffer, also the main details of Meixner’s method and results are included in Chapter 1. Finally, it is briefly mentioned that W. A. Al-Salam extended the results of Meixner.
In Chapter 2, several of the many applications of the classical orthogonal polynomials are discussed. These applications include first-order differential equations that characterize linear generating functions, additional first-order differential equations, second-order differential equations which lead to a way to solve the time-dependent Schrödinger equation, difference equations and numerical integration (Gaussian quadrature). Each of the applications is first developed in a general context and then examples using specific Sheffer sequences, such as the Laguerre, Hermite, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials, are provided.
In 1939, Sheffer also briefly described how the generating function characterizing the Type 0 polynomial sequences can also be extended to the case of arbitrary B-Type \(k\) as follows: \[ A(t) \exp{[ x H_1(t) + \dotsb + x^{k+1} H_{k+1}(t) ]} = \sum_{n=0}^{\infty} P_n(x) t^n , \] with \[ H_i(t) = h_{i,i} t^i + h_{i,i+1} t^{i+1} + \dots,\;h_{1,1} \not= 0,\;i = 1,2,\dotsc,k+1. \] So far, no results have been published that specifically analyze the higher-order Sheffer classes (\(k \geq 1\) above). Therefore, in Chapter 3, a preliminary analysis of a special case of the B-Type 1 (\(k=1\)) class is presented, in order to determine which sets, if any, are also orthogonal. This part of the monograph contains thus the novel results. Moreover, the method applied in this chapter is quite useful, as it can be applied to other characterization problems as well. Furthermore, the chapter demonstrates how computer algebra packages, like Mathematica, can play an important role in the development of rigorous results in the study of orthogonal polynomials and special functions. Finally, some future research problems that can be solved using the techniques of the final chapter are discussed.