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Canonical Polynomial Sequences: Inverse Pairs 3 CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS PHILIP FEINSILVER Abstract. For V (z), analytic in a neighborhood of 0 C, ∈ V (0) = 0, V ′(0) = 0, there is an associated sequence of poly- nomials, canonical6 polynomials, that is a generalized Appell sequence with lowering operator V (d/dx). Correspondingly, the inverse function Z(v) has an associated canonical poly- nomial sequence. The coefficients of the two sequences form two mutually inverse infinite matrices. We detail the op- erator calculus for the pair of systems, illustrating the du- ality of operators and variables between them. Examples are presented including a connection between Gegenbauer and Bessel polynomials. Touchard polynomials appear as well. Alternative settings would include an umbral calculus approach as well as the Riordan group. Here we use the approach through operator calculus. Keywords: Riordan group, special functions, orthogonal polynomi- als, operator calculus, Heisenberg-Weyl algebra, Bessel operator Subject Classification: Primary 33C45, 05A40; Secondary 42C05, 17B30 arXiv:2106.02132v1 [math.FA] 3 Jun 2021 1. Introduction Our goal is to study certain families of polynomials, canonical polyno- mial systems, which are a type of generalized Appell systems. The term canonical comes from the phrase “canonical commutation relations” re- ferring, in our case, to operators acting on polynomials satisfying the boson commutation relation(s), i.e., operators whose commutator is the identity. We will present these in explicit form. Acting on polynomials in x, define the operators d D = and x = “multiplication by x” dx 2 PHILIPFEINSILVER the symbol x playing the dual rˆole of variable and multiplication oper- ator. They satisfy commutation relations [D, x]= I, where I, the identity op- erator, commutes with both D and x. Abstractly, the Heisenberg-Weyl algebra is the associative algebra generated by operators A,B,C sat- isfying the commutation relations { } [A, B]= C, [A, C] = [B,C]=0 . The standard HW algebra is the one generated by the realization D,x,I . So in this work, we are studying particular representations of{ the HW} algebra acting on spaces of polynomials. A system of polynomials pn(x) n 0 is an Appell system if it is a basis for a representation of the{ standard} ≥ HW algebra with the following properties: (1) pn is of degree n in x (2) Dpn = npn 1. − Such sequences of polynomials are canonical polynomial systems in the sense that they provide a polynomial basis for a representation of the Heisenberg-Weyl algebra, in realizations different from the standard one. Our approach is to use operator calculus methods for studying such systems. More specifically, take an analytic function V (z) defined in a neighbor- hood of 0 in C, with V (0) = 0, V ′(0) = 0. Denote W (z)=1/V ′(z) and Z(v) the inverse function, i.e., V (Z6(v)) = v, Z(V (z)) = z. Now, V (D) is defined by power series as an operator on polynomials in x. We have the commutation relations (see below, Proposition 2.1) [V (D),X]= V ′(D) , [V (D), xW (D)] = I. In other words, V = V (D) and Y = xW (D) generate a representation of the HW algebra on polynomials in x. The basis for the represen- n tation is pn(x) = Y 1. The constant function 1 = p0 is referred to as a “vacuum state”, as it is annihilated by the lowering operator V : V (D)1 = V (0) = 0. We generate the polynomial basis recursively: pn+1 = Ypn , V (D)pn = npn 1 (1) − CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 3 That is, Y is the raising operator and V (D) is the corresponding low- ering operator. Thus, pn n 0 form a system of canonical polynomials. The operator of multiplication{ } ≥ by x is given by 1 x = YV ′(D)= YZ′(V )− which is a recursion operator for the system. Our approach holds as well for multivariate systems, some examples of which will appear in the final sections of this work. Primarily this work is a source providing useful examples of such se- quences with details about the operators/representations involved. Many well-known sequences of polynomials, or closely-related families, will make their appearance. 2. Dual vector fields and canonical polynomial sequences Identifying vector fields with first-order partial differential operators, consider a variable A with corresponding partial differential operator ∂A. Given V as above, let Y˜ be the vector field Y˜ = W (A) ∂A. An op- erator function of D acts on eAx as multiplication by the corresponding function in the variable A. The basic fact is the following primitive version of the Fourier trans- form: Ax Ax Ax xW (D) e = W (A)∂A e = xW (A) e (2) where D denotes d/dx, and W is a function analytic in a neighborhood of 0 C. ∈ What this does is exchange the operators in the (x, D) variables with corresponding operators in (A, ∂A) variables. Thus each vector field has its dual and vice versa. That is, xW (D) is the dual vector field corresponding to the vector field W (A)∂A. We abbreviate “dual vector field” by “dvf”. When discussing functions of D, we use the variable z for complex variables defining the function involved. For an operator f(D), the function f(z) may be referred to as its symbol. We have the commu- tation relation, Dx xD = I, x playing the dual rˆole of variable and − 4 PHILIPFEINSILVER operator of multiplication by x. When there is no ambiguity of mean- ing, we write as well [z, x]= I. We have the “canonical function” V (z) holomorphic in a neighborhood of 0 C, with V (0) = 0, and the derivative V ′(0) = 0. And the func- tional∈ inverse of V , denoted by Z(v). 6 The associated dvf is called a “canonical variable” and is given by Y = xW (D) 1 where W (z)= V ′(z)− is the reciprocal of the derivative V ′(z). We remark the facilitating feature of this operator calculus that any function of D expressible as a series expansion, perhaps formal, acts finitely on polynomials. Defining the action on exponentials as evalua- tion gives a well-defined calculus acting on polynomials, exponentials, and linear combinations of exponentials with polynomial coefficients. Cf. [3, 6]. Proposition 2.1. The commutation relation [V (D), x]= V ′(D) the derivative of V . Proof. We use the approach of equation(2) and the usual product rule: Ax Ax Ax V (D)x e = V (D)∂A e = ∂AV (A) e Ax Ax Ax Ax = V ′(A) e + V (A)x e = V ′(D) e + xV (D) e so that V (D)x xV (D)= V ′(D) − acting on eAx. Then differentiating repeatedly with respect to A yields the result for polynomials as well. Proposition 2.2. The commutation relations [V (D),Y ]= I hold. Proof. By the above Proposition: [V (D), xW (D)] = V ′(D)W (D)= I as required. CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 5 Notation. We denote the operator V (D) by . V In the (Y, ) calculus, parallel to the usual (x, D) calculus, Y takes the rˆole of multiplicationV by x and the rˆole of differentiation. V Remark. Note that W is essentially the derivative of the inverse function Z(v). That is, Z(V (z)) = z = Z′(V (z))V ′(z)=1= Z′(V (z)) = W (z) ⇒ ⇒ and substituting z = Z(v), W (Z(v)) = Z′(v) relations well-known from calculus. Example. For the Poisson case, we take V (z) = ez 1, so that z − W (z)= e− and Z(v) = log(1 + v). 2.1. Canonical polynomials. Given V , we define the corresponding canonical polynomials, p (x) by { n } p (x)=(xW (D)) (xW (D))1 = Y n1 n ··· so that pn satisfy the relations Ypn = pn+1 pn = npn 1 V − with p0 = 1, p0 = 0. Because of this property, is called the lowering operator or velocityV operator. V Definition. Given a canonical polynomial sequence we write k pn(x)= pnkx Xk We call the infinite matrix pnk , the matrix of coefficients or simply refer to the coefficients of the{ polynomial} sequence. We can refer as well to the P -matrix of the system. Observe (e.g., setting x = 0 in the generating function) that pn(0) = 0 for all n> 0. Remark. One can work directly with the coefficients pnk , as infinite matrices, as basic objects of study. Cf. [7, 9, 10]. { } 6 PHILIPFEINSILVER z Example. For the Poisson case, W (z)= e− acts on functions of x as a shift operator W (D) f(x)= f(x 1) − so Y f(x)= xf(x 1) and iterating, starting from p = 1, yields − 0 p (x)= x(x 1) (x (n 1)) = x(n) n − ··· − − falling factorial polynomial. One checks directly that = eD 1 acts accordingly. V − We continue by looking at several basic examples and structures asso- ciated to sequences of canonical polynomials. 2.2. Flow of a dual vector field. We will now derive the exponen- tial generating function for the sequence pn . From the connection between vector fields and dvf’s, observe that{ } we are interested in the flow of the dual vector field. We would like to calculate the solution to ∂u = Yu, u(0) = f(x) ∂t for the dvf Y = xW (D). Let Y˜ = W (A)∂A denote the corresponding vector field. Using equa- tion (2), we have etY eAx = etY˜ eAx by, say, applying Y iteratively on the left side, and on the right side moving Y past powers of Y˜ and converting to Y˜ when acting on eAx. We know how to calculate the flow of the vector field Y˜ by the charac- teristic equations A˙ = W (A) Multiplying both sides by V ′(A) we get AV˙ ′(A)=1 Now the left-hand side is an exact derivative.
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