Canonical Polynomial Sequences: Inverse Pairs 3

Home , Polynomial sequence

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They satisfy commutation relations [D, x]= I, where I, the identity operator, commutes with both D and x. Abstractly, the Heisenberg-Weyl algebra is the associative algebra generated by operators A,B,C satisfying 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 neighborhood 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 lowering 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 sequences 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 operator 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 commutation 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 functional∈ 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 rôle of multiplicationV by x and the rôle 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 associated to sequences of canonical polynomials. 2.2. Flow of a dual vector field. We will now derive the exponential 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 equation (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. I.e., d V (A(t))=1 dt Integrating, with initial conditions A(0) = A, we get V (A(t)) = V (A)+ t Solving, we have A(t)= Z(t + V (A)) CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 7

Thus, etY eAx = etY˜ eAx = exZ(t+V (A)) (3) As a corollary we have the action of the dvf Y on the vacuum function equal to 1 by setting A = 0: etY 1= exZ(t) In our context, replacing t by v we have Theorem 2.3. Main Formula For a canonical function V (z), with W (z)=1/V ′(z), Z(V (z)) = z, let Y = xW (D) be the associated canonical variable. Then we have evY eAx = exZ(v+V (A)) And, in particular, evY 1= exZ(v) . We have the expansion vn evY 1= exZ(v) = p (x) n! n n 0 X≥ z D Example. For the Poisson case: V (z)= e 1, Y = xe− . We found above that − p (x)= Y n1= x(x 1) (x n +1) = x(n) n − ··· − With Z(v) = log(1 + v), the expansion is vn x (1 + v)x = x(n) = vn n! n n 0 n 0 X≥ X≥ the standard binomial theorem.

2.3. Semigroup and group properties. Observing that the generating function is an exponential, Proposition 2.4. We have the group property, binomial type relations, cf. [4, 5]: n pn j(s)pj(t)= pn(s + t) . j − j X which follows upon expanding esZ(v)etZ(v) = e(s+t)Z(v) and regrouping. We have as well relations for the coeﬃcients p : { nk} 8 PHILIPFEINSILVER

Proposition 2.5. The coeﬃcients satisfy the semigroup property: n k + l pn j,k pjl = pn,k+l . j − k j X Proof. Expand the generating function two ways: 1 vn exZ(v) = xk Z(v)k = p xk k! n! nk n Xk X Xk and see that vn k! p Z(v)k = nk n! n X Now use the semigroup property Z(v)kZ(v)l = Z(v)k+l and rearrange factorials to arrive at the result. 2.4. Composition of systems. Riordan group. Umbral composition. Note. For simplicity, the range of summations is suppressed in these calculations. Generically n 0 running to with k running from 0 to n. ≥ ∞

Start with two systems vn exZ1(v) = p(1)xk = evY1 1 n! nk X vn X exZ2(v) = p(2)xk = evY2 1 n! nk When composing these systems,X weX will introduce scaling factors for additional ﬂexibility. Consider the composed P -matrices and resum (with γ and σ as scale factors): vn (vγ)n γn p(2) σm p(1) xk = p(2) σm Y m1 n! nm mk n! nm 1 X XX = Xe(σY1)Z2(γv)1X with σY1 playing the rˆole of x for the Z2 system. Now consider σZ2(γv) as the v-variable for the Z1 system: (σZ2(γv))Y1 e 1 = exp xZ1(σZ2(γv)) . That is, Theorem 2.6. Composition Theorem We have the expansion vn exp xZ (σZ (γv)) = p(12)(x) 1 2 n! n X CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 9 where (12) n (2) m (1) k pn (x)= γ pnmσ pmkx . XX Call V12 the canonical operator for the combined system. Then we can solve: 1 1 z = Z σZ (γV ) = V = V V (z) 1 2 12 ⇒ 12 γ 2 σ 1 Remark. 1. The construction for polynomial f: f(x)= c xm c p (x)= f(Y )1 m → m m is umbral compositionX. Cf. [4, 5]. X

2. Composing the systems by working with the P -matrices directly is the technique of the Riordan group, [7].

Our approach uses operators and variables explicitly.

2.4.1. Poisson subordination. A common situation arises where v (2) Z2(v) = e 1. Then pnm = S(n, m), Stirling numbers of the second kind. We− call this “Poisson subordination by . . . ” — whatever the type of Z is. Consider Z given, and Z = ev 1. We have 1 1 2 − vn exZ2(v) = p(2)(x) where p(2)(x)= S(n, m)xm n! n n and by the aboveX composition theorem we have, withX Z(v)= Z (σ(eγv 1)), 1 − n m (1) pn(x)= γ S(n, m)σ pm (x) . X

3. Inverse Pairs

Start with V (z). We have Y = xW (D) and polynomials pn(x) with generating function vn evY 1= exZ(v) = p (x) n! n n 0 X≥ with operators acting on functions of x, in particular z corresponds to D = d/dx. 10 PHILIP FEINSILVER

Now consider the inverse Z(v) as the canonical operator for its own system of polynomials in the variable y. Then form Z′(v) and the (dual) canonical operator

1 X = yZ′(∂y)−

Here we correspond v to ∂y = d/dy. Now operators act on functions of y and we have the corresponding canonical polynomial sequence: zn ezX 1= eyV (z) = q (y) n! n n 0 X≥ with Q-matrix q : { nk} k qn(y)= qnky satisfying X

Xqn = qn+1

qn = nqn 1 Z − with = Z(d/dy). Z

Apply the composition theorem with Z1 = Z and Z2 = V , σ = γ = 1. We get vn vn exZ1(Z2(v)) = exv = q p (x)= q p xk n! nm m n! nm mk X XX X XX as expected, we have reciprocal relations, umbral composition,

n qnmpm(x)= x = qn(Y )1 and the P and Q matricesX inverse to each other:

qnmpmk = δnk

And dually, X

n pnmqm(y)= y = pn(X)1

We identify z with d/dxXand v with d/dy, with corresponding operators for capitalized variables.

We thus have: CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 11

d d d d z v ∼ dx Z ∼ dX ∼ dy V ∼ dY

V (z) Z(v)

V ′(z) Z′(v)

1 1 Y = xV ′(z)− = xZ′( ) X = yZ′(v)− = yV ′( ) V Z 1 1 x = YV ′(z)= YZ′( )− y = XZ′(v)= XV ′( )− V Z

Ypn = pn+1 Xqn = qn+1

pn = npn 1 qn = nqn 1 V − Z −

n n pn(X)1 = y qn(Y )1 = x

vn zn evY 1= exZ(v) = p (x) ezX 1= eyV (z) = q (y) n! n n! n n 0 n 0 X≥ X≥ Table 1. Generic system

3.1. Commutation Relations. The commutation relations are: [z, x]= I and [ ,X]= I Z [v,y]= I and [ ,Y ]= I V These are four sets of canonical pairs. The two reciprocal systems are inverse pairs. Here is a summary illustrating the principal constructions, showing the duality between the systems of inverse pairs. From one side to the other, capitals are replaced by lower case and vice versa.

1 Note that the formulation x = YZ′( )− gives the action of multipli- V cation by x on the polynomial sequence pn . Similarly for the dual system in terms of y and the (inverse) sequence{ } q . { n} We proceed to study some examples in detail. For each system, we find the main operators involved. Some of the p- and q-polynomials are presented, as the coefficients arising in these systems often have interesting combinatorial significance (which, however, we will not discuss here). 12 PHILIP FEINSILVER

4. Poisson-Stirling system The name Poisson comes from the moment generating function for a Poisson distribution with parameter y:

n y zn y y(ez 1) e− e = e − n! n 0 X≥ We have for canonical variables

V (z)= ez 1 Z(v) = log(1 + v) − z 1 V ′(z)= e Z′(v)= 1+ v z 1 Y = xe− = x(1 + )− X = y(1 + v)= yeZ V z 1 x = Y e = Y (1 + ) y = X = Xe−Z V 1+ v The generating functions are vn evY 1=(1+ v)x = p (x) n! n n 0 X≥ and n zX y(ez 1) z e 1= e − = q (y) . n! n n 0 X≥ Proposition 4.1. We have

p (x)= x(x 1) (x (n 1)) = s(n, k) xk n − ··· − − q (y)= S(n, k) yk = (y) X n Tn with s(n, k) and SX(n, k) denoting Stirling numbers of the ﬁrst and second kinds respectively, cf. [1]. The polynomials q are Touchard polynomials, which we denote by . n T Proof. For the q-polynomials, use the formula (ez 1)m zn − = S(n, m) . m! n! n X

Selection of each: CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 13

p0 =1 p1 = x p = x2 x 2 − p = x3 3x2 +2x 3 − p = x4 6x3 + 11x2 6x 4 − − p = x5 10x4 + 35x3 50x2 + 24x 5 − − p = x6 15x5 + 85x4 225x3 + 274x2 120x 6 − − − p = x7 21x6 + 175x5 735x4 + 1624x3 1764x2 + 720x 7 − − − p = x8 28x7 + 322x6 1960x5 + 6769x4 13132x3 + 13068x2 5040x 8 − − − −

q0 =1

q1 = y 2 q2 = y + y 3 2 q3 = y +3y + y 4 3 2 q4 = y +6y +7y + y 5 4 3 2 q5 = y + 10y + 25y + 15y + y 6 5 4 3 2 q6 = y + 15y + 65y + 90y + 31y + y 7 6 5 4 3 2 q7 = y + 21y + 140y + 350y + 301y + 63y + y 8 7 6 5 4 3 2 q8 = y + 28y + 266y + 1050y + 1701y + 966y + 127y + y

4.1. Recurrences. We have the standard actions Ypn = pn+1 and pn = npn 1 and correspondingly for X and acting on the q- Vpolynomials.− For example, X = y(1 + v)= y + yvZgives the relation dq q = yq + y n . n+1 n dy On the polynomials p , the formula x = Y (1 + )= Y + Y reads { n} V V xpn = pn+1 + npn so p =(x n)p n+1 − n 14 PHILIP FEINSILVER verifying the formula found above for pn. While the relation y = Xe−Z gives a recurrence as the expansion n yqn = qn+1 nqn + qn 1 − 2 − −··· We derive this using the fact that e−Z acts as a shift operator on functions of X: e−Z f(X)1 = f(X 1)1 − so n n k yqn = X(X 1) 1= ( 1) qn+1 k − k − − Xk as stated above. And we note that the semigroup properties holding in general for the polynomials’ coeﬃcients give convolution relations for the Stirling numbers of the ﬁrst and second kinds.

5. Laguerre system We have for canonical variables z v V (z)= Z(v)= 1 z 1+ v − 1 1 V ′(z)= Z′(v)= (1 z)2 (1 + v)2 −2 2 2 2 Y = x(1 z) = x(1 + )− X = y(1 + v) = y(1 )− − V − Z

2 2 2 2 x = Y (1 z)− = Y (1 + ) y = X(1 + v)− = X(1 ) − V − Z The generating functions are x v vn evY 1= e 1+ v = p (x) n! n n 0 X≥ and y z zn ezX 1= e 1 z = q (y) − n! n n 0 X≥ We use the notation from online dlmf.nist.gov, [1, 18.5.12]. Proposition 5.1. We have n ( 1) p (x)=( 1) n! L − (x) n − n n n Γ(n) k ( 1) q (y)= y = n! L − ( y) n k Γ(k) n − Xk=1 CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 15

The polynomials qn are special Laguerre polynomials. Selection of each:

p0 =1 p1 = x p = x2 2x 2 − p = x3 6x2 +6x 3 − p = x4 12x3 + 36x2 24x 4 − − p = x5 20x4 + 120x3 240x2 + 120x 5 − − p = x6 30x5 + 300x4 1200x3 + 1800x2 720x 6 − − − p = x7 42x6 + 630x5 4200x4 + 12600x3 15120x2 + 5040x 7 − − − p = x8 56x7 + 1176x6 11760x5 + 58800x4 141120x3 + 141120x2 40320x 8 − − − −

q0 =1 q1 = y 2 q2 = y +2y 3 2 q3 = y +6y +6y 4 3 2 q4 = y + 12y + 36y + 24y 5 4 3 2 q5 = y + 20y + 120y + 240y + 120y 6 5 4 3 2 q6 = y + 30y + 300y + 1200y + 1800y + 720y 7 6 5 4 3 2 q7 = y + 42y + 630y + 4200y + 12600y + 15120y + 5040y 8 7 6 5 4 3 2 q8 = y + 56y + 1176y + 11760y + 58800y + 141120y + 141120y + 40320y

5.1. Recurrences. From x = Y (1 + )2 = Y +2Y + Y 2 we get V V V xpn = pn+1 +2npn + n(n 1)pn 1 − − or

pn+1 =(x 2n)pn n(n 1)pn 1 − − − − Similarly, y = X 2X + X 2 gives − Z Z qn+1 =(y +2n)qn n(n 1)qn 1 − − − 16 PHILIP FEINSILVER while Y = x(1 z)2 gives the diﬀerential recurrence − p = x(p 2p′ + p′′ ) . n+1 n − n n

6. Hermite-Bessel system Here we are looking at polynomials related to the Gaussian distribution. We have for canonical variables V (z)= z z2/2 Z(v)=1 √1 2v − − − 1 V ′(z)=1 z Z′(v)= − √1 2v 1 1/2 − Y = x(1 z)− = x(1 2 )− X = y√1 2v = y(1 ) − − V − − Z 1 1 x = Y (1 z)= Y (1 2 )1/2 y = X = X − − V √1 2v 1 − − Z The generating functions are n vY x(1 √1 2v) v e 1= e − − = p (x) n! n n 0 X≥ and n zX y(z z2/2) z e 1= e − = q (y) n! n n 0 X≥ For the modiﬁed Hermite polynomials, see [1, 18.12.16]. For Bessel polynomials, cf. [1, 18.34.2]. Proposition 6.1. We have n k Γ(n + k) x − pn(x)= xθn 1(x)= − Γ(n k) k! 2k X − n (2k)! k n k q (y)= He (y,y)= ( 1) y − n n 2k 2k k! − X where θn are Bessel polynomials and Hen(x, t) are probabilists’ Hermite polynomials (orthogonal with respect to a Gaussian distribution with mean zero and variance t) with generating function n xz z2t/2 z e − = He (x, t) (4) n! n n 0 X≥ For Bessel polynomials, the formulation n 1 − n + k 1 (2k)! n k xθn 1(x)= − x − − 2k 2k k! Xk=0 CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 17 shows clearly the connection with moments of the Gaussian distribution or, in this case, one should say moments of the χ2 distribution. − Selection of each:

p0 =1 p1 = x 2 p2 = x + x 3 2 p3 = x +3x +3x 4 3 2 p4 = x +6x + 15x + 15x 5 4 3 2 p5 = x + 10x + 45x + 105x + 105x 6 5 4 3 2 p6 = x + 15x + 105x + 420x + 945x + 945x 7 6 5 4 3 2 p7 = x + 21x + 210x + 1260x + 4725x + 10395x + 10395x 8 7 6 5 4 3 2 p8 = x + 28x + 378x + 3150x + 17325x + 62370x + 135135x + 135135x

q0 =1

q1 = y q = y2 y 2 − q = y3 3y2 3 − q = y4 6y3 +3y2 4 − q = y5 10y4 + 15y3 5 − q = y6 15y5 + 45y4 15y3 6 − − q = y7 21y6 + 105y5 105y4 7 − − q = y8 28y7 + 210y6 420y5 + 105y4 8 − −

Notice the diagonals along the Hermite polynomials yield the coeﬃ- cients for the Bessel polynomials up to sign. 18 PHILIP FEINSILVER

6.1. Recurrences. With x = Y (1 2 )1/2 we calculate − V x2 = Y (1 2 )1/2Y (1 2 )1/2 − V − V d = Y 2(1 2 )+ Y ( (1 2 )1/2)(1 2 )1/2 − V d − V − V V = Y 2(1 2 ) Y − V − Hence, x2p = p (2n + 1)p n n+2 − n+1 or, shifting the index n back by one, and rearranging,

2 pn+1 = (2n 1)pn + x pn 1 − − a recurrence formula for Bessel polynomials. For Hermite polynomials, we have X = y y yielding − Z

qn+1 = yqn ynqn 1 = y(qn nqn 1) . − − − −

7. Arcsinh system

V (z) = arcsinh z Z(v) = sinh v 1 V ′(z)= Z′(v) = cosh v √1+ z2 1 Y = x√1+ z2 = x cosh X = y sech v = y V √1+ 2 1 Z x = Y = Y sech y = X cosh v = X√1+ 2 √1+ z2 V Z The generating functions are vn evY 1= ex sinh v = p (x) n! n n 0 X≥ and zn ezX 1= ey arcsinh z = q (y) n! n n 0 X≥ CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 19

Proposition 7.1. We have, denoting the Touchard polynomials (x), Tn

n j pn(x)= ( 1) n j(x/2) j( x/2) j − T − T − j X (n 3)/2 y − (y2 (2k + 1)2) , n odd  k=0 − qn(y)=  Q (n 2)/2  − (y2 (2k)2) , n even k=0 −   Q  v v Proof. For pn , write sinh v =(e e− )/2 and use the Poisson-Stirling q-polynomials:{ } −

x sinh v x(ev 1)/2 x(e−v 1)/2 e = e − e− − vn ( 1)nvn = (x/2) − ( x/2) n! Tn n! Tn − n ! n ! X X which yields the result upon rearrangement. The formula for the qn will be seen directly from the recurrence derived below.

Selection of each:

p0 =1

p1 = x 2 p2 = x 3 p3 = x + x 4 2 p4 = x +4x 5 3 p5 = x + 10x + x 6 4 2 p6 = x + 20x + 16x 7 5 3 p7 = x + 35x + 91x + x 8 6 4 2 p8 = x + 56x + 336x + 64x 20 PHILIP FEINSILVER

q0 =1

q1 = y 2 q2 = y q = y3 y 3 − q = y4 4y2 4 − q = y5 10y3 +9y 5 − q = y6 20y4 + 64y2 6 − q = y7 35y5 + 259y3 225y 7 − − q = y8 56y6 + 784y4 2304y2 8 − −

7.1. Recurrences. Bessel operator. Here we have the interesting appearance of the Bessel operator. Eﬀectively, Y is the square root of the Bessel operator (xD)2 + x2. To see this, calculate, with Y = x√1+ z2, z = D = d/dx,

Y 2 = x√1+ z2 x√1+ z2 d = x2(1 + z2)+ x (√1+ z2) √1+ z2 dz = x2(1 + z2)+ xz = x2 +(xz)2 .

Thus, the polynomials pn are formed by repeated application of the Bessel operator. The diﬀerential recurrence has the form:

d 2 p = x2p + x p n+2 n dx n For the q-sequence, use the dual relation y = X√1+ 2 to get similarly Z y2 = X2 +(X )2 Z That is, 2 2 y qn = qn+2 + n qn or q =(y2 n2)q n+2 − n and hence the formulas quoted above. CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 21

8. Tanh system

V (z) = tanh z Z(v) = arctanh v

2 1 V ′(z) = sech z Z′(v)= 1 v2 1 − Y = x cosh2 z = x X = y(1 v2)= y sech2 1 2 − Z −V 1 x = Y sech2z = Y (1 2) y = X = X cosh2 −V 1 v2 Z − and the generating functions are vn evY 1= ex arctanh v = p (x) n! n n 0 X≥ zn ezX 1= ey tanh z = q (y) n! n n 0 X≥ Proposition 8.1. We have the “zero-step” Krawtchouk polynomials n x (n k) x (k) p (x)= ( 1)k − =( 1)n K (x/2;1/2, 0) n k − 2 − 2 − n Xk cf. [1, 18.23.3], dropping the factor involving N. We have a(n) = a(a 1) (a n + 1) denoting falling factorial (factorial power) − ··· − and for the q-polynomials we have Poisson subordination by Laguerre: n m Laguerre qn(y)= S(n, m)2 − pm (y) . m X Proof. Recall e2z 1 1 1+ v tanh z = − and arctanh v = log e2z 1 2 1 v − − So for p-polynomials 1+ v x/2 vn = p (x) . 1 v n! n n − X Expanding by the binomial theorem and combining terms yields the formula.

For the q-polynomials, refer to Section §2.4.1. We verify Poisson subordination with Z1(v)= v/(1 + v), σ =1/2, γ = 2. Here we want the 22 PHILIP FEINSILVER expansion in powers of z. We have v = (1/2)(e2z 1) and, exchanging Z and V : − 1 e2z 1 e2z 1 V = − = − = tanh z 2 1+(1/2)(e2z 1) e2z +1 − thus the result for the q-polynomials. Selection of each:

p0 =1

p1 = x 2 p2 = x 3 p3 = x +2x 4 2 p4 = x +8x 5 3 p5 = x + 20x + 24x 6 4 2 p6 = x + 40x + 184x 7 5 3 p7 = x + 70x + 784x + 720x 8 6 4 2 p8 = x + 112x + 2464x + 8448x

q0 =1

q1 = y 2 q2 = y q = y3 2y 3 − q = y4 8y2 4 − q = y5 20y3 + 16y 5 − q = y6 40y4 + 136y2 6 − q = y7 70y5 + 616y3 272y 7 − − q = y8 112y6 + 2016y4 3968y2 8 − −

8.1. Recurrences. With x = Y (1 2) we have directly −V xpn = pn+1 n(n 1)pn 1 − − − and the recurrence

pn+1 = xpn + n(n 1)pn 1 . − − CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 23

Duallly, we have X = y(1 v2) yielding the diﬀerential recurrence − q = yq yq′′ = y(q q′′) . n+1 n − n n − n 2 2 2 2 From y = X cosh , writing cosh = e Z + e− Z +2 /4, we have Z Z X yq = (X + 2)n +(X 2)n +2Xn 1 n 4 − X n n k k k 1 = X − (2 +( 2) ) 1+ qn+1 4 " k − # 2 Xk 1 n n+1 2j 2j+1 1 = X − 2 1+ q 4 2j 2 n+1 j X or, taking out the term j = 0,

n 2j 1 yqn = 2 − qn+1 2j + qn+1 2j − j 1 X≥ and ﬁnally,

n 2j 1 qn+1 = yqn 2 − qn+1 2j . − 2j − j 1 X≥ 9. Gegenbauer system

V (z)= log(1 2αz + z2) Z(v)= α √α2 + e v 1 − − − − − v 2α 2z e− V ′(z)= − Z′(v)= 1 2αz + z2 2√α2 + e v 1 − − −

2 1 2αz + z e−V v 2 v 2α 2 Y = x − = x X =2ye √α + e− 1= y − Z 2α 2z 2√α2 + e 1 − 1 2α + 2 − −V − − Z Z

v 2 2α 2z 2 e− 1 2α + x = Y − =2Y eV α + e 1 y = X = X − Z Z 2 −V 2 v 1 2αz + z − 2√α + e− 1 2α 2 − p − − Z and the generating functions are vn evY 1= exZ(v) = p (x) n! n n 0 X≥ n zX yV (z) 2 y z e 1= e = (1 2αz + z )− = q (y) − n! n n 0 X≥ 24 PHILIP FEINSILVER

Proposition 9.1.

n k 2 k Bessel p (x)= S(n, k)( 1) − (2α )− p (αx) n − k Xk Bessel where pk are the Bessel polynomials xθn 1(x) as in the Hermite- Bessel system. That is, these arise as Poisson− subordination by Bessel, with scaling x as well. And

(x) qn(x)= n! Cn (α)

Gegenbauer polynomials in the variable α with parameter x.

Proof. For the form of the p-polynomials, we verify Poisson subordina- 2 1 tion with Z1(v)=1 √1 2v, σ = (2α )− , γ = 1. So we have, scaling by α, − − − −

1 v Z = α 1 1 2 − 2 (e− 1) − r − 2α − ! in agreement with Z as given above.

Selection of each:

p0 =1 x p = 1 2α 1 1 x2 p = x + + 2 −2α 4α3 4α2 3 3 1 3 3 x3 p = x2 + + x + + 3 −4α2 8α4 2α − 4α3 8α5 8α3 3 3 7 9 15 1 7 9 15 p = x3 + + x2 + + x + + 4 −4α3 8α5 4α2 − 4α4 16α6 −2α 4α3 − 4α5 16α7 x4 + 16α4 5 5 25 15 45 15 75 75 105 p = x4 + + x3 + + x2 + + 5 −8α4 16α6 8α3 − 4α5 32α7 −4α2 8α4 − 8α6 32α8 1 15 75 75 105 x5 + x + + + 2α − 4α3 8α5 − 8α7 32α9 32α5 CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 25 q0 =1 q1 =2αy q =4α2y2 + y 4α2 2 2 − q =8α3y3 + y2 24α3 12α + y 16α3 12α 3 − − q = 16α4y4 + y3 96α4 48α2 + y2 176α4 144α2 + 12 + y 96α4 96α2 + 12 4 − − − q = 32α5y5 + y4 320α5 160α3 + y3 1120α5 960α3 + 120α 5 − − + y2 1600α5 1760α3 + 360α + y 768α5 960α3 + 240α − − 9.1. Recurrence. From 1 2α + 2 y = X − Z Z 2α 2 − Z we have y(2α 2 )= X(1 2α + 2) − Z − Z Z Applying this relation to qn yields

y(2αqn 2nqn 1)= qn+1 2αnqn + n(n 1)qn 1 − − − − − and rearranging, we have

qn+1 =2α(y + n)qn n(2y + n 1)qn 1 . − − −

10. Multivariate Case We indicate how the multivariate case works. For N > 0, V (z) = (V1(z),...,VN (z)), z =(z1,...,zN ), with Z(V (z)) = z, the functional inverse. V (0) = 0 CN , holomorphic in a neighborhood of the origin. ∈ ∂Vi Now V ′(z) is the Jacobian , with inverse matrix W (z). The ∂zj canonical variables are Yj = xλWλj(D) with repeated Greek indices summed. One checks the commutation relations

[Vi(D),Yj]= δij I

[Yi,Yj]=0 The canonical polynomials are p (x)= Y n1 Y nN 1 n 1 ··· N 26 PHILIP FEINSILVER with multi-index n =(n1,...,nN ). The generating function is n x Z(v) v e · = p (x) n! n n X with x Z(v)= j xjZj(v), dot product, and the usual conventions for multi-indices.· P We illustrate with a multivariate version of the Hermite-Bessel system.

1 2 Example. Let Vj(z)= zj 2 zi . Then we ﬁnd Zj(v)= vj + S(v) where − P 1 S(v)= 1 v (1 v )2 N v2 N − j − − j − j q The Jacobian is X X X V ′ = I Z − where Z is the rank one matrix with every row equal to (z1,...,zN ). And 1 1 W =(V ′)− = I + Z 1 zi so − DPj Y = x +( x ) . j j i 1 D − i The p-polynomials are a family ofP multivariate Bessel polynomials. We can ﬁnd the q-polynomials expressed as productsP of Hermite polynomials. Note 1 y V (z)= y z z2 · j j − 2 i X 1 X = y z y z2 j j − 2 i j X 1P X = y z y z2 j j − 2 i j So X P

y V 1 2 e · = exp y z ( y )z j j − 2 i j j Y nj P zj = Henj (yj, yi) nj! j nj Y X P zn = He (y , y ) n! nj j i n n=(n1,...,nN ) X Y P CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 27 cf. equation (4).

10.1. Linear maps and symmetric tensor powers. An interesting special case arises when V is a linear map. Let A be a given N N matrix. Form × 1 T V (z)=(A− ) z 1 T T with T denoting transpose. We have V ′ =(A− ) and W = A . So

Yj = xλWλj = xλAjλ =(Ax)j i.e. Y = Ax.

T T 1 And Z(v) = A v, with Z′ = A and X = A− y. The generating function is n x AT v Ax v v m e · = e · = A¯ x n! nm n m That is, X X n n m Y 1=(Ax) = A¯nmx For given homogeneous degree d, theX map A A¯ is a multiplicative map taking the space of homogeneous polynomials→ of degree d to itself. For a matrix group, for each d, this gives a group representation, the action of the group on polynomials, alternatively, action on the space of symmetrized tensors (of the underlying vector space), cf. [8].

10.1.1. Homorphism property. The composition theorem here becomes the property that the map A A¯ is a homomorphism. Take two → T T matrices A and B and form Z1 = B v, Z2 = A v. Then T T T Z1(Z2(v)) = B A v =(AB) v And the exponential vn vn exZ1(Z2(v)) = A¯ B¯ xm = AB xm n! nk km n! nm as expected. X X X X 10.1.2. Lie algebra map. We can also study the induced map at the Lie algebra level. Namely, deﬁne Γ(A) to be the matrix satisfying exp(tA) = exp(tΓ(A)) As the generator of the image of the one-parameter group generated by A we have d Γ(A)= exp(tA) . dt t=0

28 PHILIP FEINSILVER

Proposition 10.1. The generating function for the matrices Γ(A) is given by n x v v m (Ax v)e · = Γ(A) x · n! nm n X X Proof. Start with vn exp (etAx) v = (etA) xm · n! nm Now diﬀerentiate with respect to Xt and setXt = 0. We get n tA tA x v v m (Ae x) v exp((e x) v) =(Ax v)e · = Γ(A) x · · t=0 · n! nm as stated. X X

This map plays an important part in certain multivariate orthogonal systems, see, e.g., [2].

11. Concluding remarks One can work in the somewhat more general setting of Sheffer sequences. From our perspective a natural way to approach these systems is as canonical systems evolving under the action of a one-parameter group. Let H(D) generate a one-parameter group acting on the polynomials p . We have { n} vn vn etH(D)exZ(v) = exZ(v)+tH(Z(v)) = etH(D)p (x) = p (x, t) . n! n n! n The relation X X tH(D) tH(D) e xe− = x + tH′(D) yields the raising operator for the new system tH(D) tH(D) = e Y e− =(x + tH′(D))W (D)= Y + tH′(D)W (D) R i.e. pn(x, t) = pn+1(x, t) with = V (D) remaining as the lowering operator,R [ , ]= I. These systemsV play a principal rôle in [4, 7]. V R References [1] Digital Library of Mathematical Functions, online, http://dlmf.nist.gov. [2] Philip Feinsilver. Krawtchouk-Griffiths systems I: matrix approach. Commun. Stoch. Anal., 10(3):Article 3, 297–320, 2016. [3] Philip Feinsilver and RenéSchott. Algebraic structures and operator calculus. Vol. III, volume 347 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. Representations of Lie groups. [4] Steven Roman. The umbral calculus, volume 111 of Pure and Applied Math- ematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. CANONICAL POLYNOMIAL SEQUENCES: INVERSE PAIRS 29

[5] Steven M. Roman and Gian-Carlo Rota. The umbral calculus. Advances in Math., 27(2):95–188, 1978. [6] Gian-Carlo Rota, D. Kahaner, and A. Odlyzko. On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl., 42:684–760, 1973. [7] Louis W. Shapiro, Seyoum Getu, Wen Jin Woan, and Leon C. Woodson. The Riordan group. Discrete Appl. Math., 34(1-3):229–239, 1991. Combinatorics and theoretical computer science (Washington, DC, 1989). [8] T. A. Springer. Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin-New York, 1977. [9] P. R. Vein. Matrices which generate families of polynomials and associated inﬁnite series. J. Math. Anal. Appl., 59(2):278–287, 1977. [10] P. R. Vein. Identities among certain triangular matrices. Linear Algebra Appl., 82:27–79, 1986.

Department of Mathematics, Southern Illinois University, Carbon- dale, IL. 62901, U.S.A.