Difference Equations for Multiple Charlier and Meixner Polynomials 1

Difference Equations for Multiple Charlier and Meixner Polynomials 1 WALTER VAN ASSCHE Department of Mathematics Katholieke Universiteit Leuven B-3001 Leuven, Belgium E-mail: walter@wis.kuleuven.ac.be Abstract We give a third order difference equation for multiple Charlier polynomials and two types of multiple Meixner polynomials. These polynomials are orthogonal with respect to two measures on the integers N. Keywords equation Discrete orthogonality, multiple orthogonal polynomials, difference AMS Subject Classification 33C45, 39A13, 42C05 1 Discrete orthogonal polynomials Suppose µ is a discrete positive measure on the real line. We will only consider discrete measures for which the support is a subset of N the linear lattice. The corresponding monic orthogonal polynomials {P n : n = 0, 1, 2,...} are such that P n is a monic polynomial of degree n for which P n xx k dµx = 0, k = 0, 1, 2,..., n 1. On the linear lattice we have two important operators, namely the forward difference operator for which fx = fx + 1 fx, and the backward difference operator for which fx = fx fx 1. If we use the Pochhammer symbol a k = aa + 1a + 2 a + k 1, with a 0 = 1, then the polynomials x k are much more convenient as building 1 Research supported by INTAS 00-272 and project G.0184.02 of FWO-Vlaanderen

blocks than the monomials x k, because of the relation x k = k x k 1. The orthogonality is therefore easier to formulate as P n x x k dµx = 0, k = 0, 1, 2,..., n 1. The classical discrete orthogonal polynomial on the linear lattice consist of the following families [5, 7, 8], where we use δ k for the Dirac measure with mass 1 at k N: The Charlier polynomials C n x; a for which a k µ = k! δ k, Poisson distribution, with a > 0. The Kravchuk polynomials K n x; p, N for which N N µ = p k 1 p N k δ k, k binomial distribution, with 0 < p < 1 and N N. The Meixner polynomials M n x; β, c for which β k c k µ = δ k, k! negative binomial distribution, with β > 0 and 0 < c < 1. The Hahn polynomials Q n x; α, β, N for which N α + k β + N k µ = δ k, k N k hypergeometric distribution, with α, β > 1 or α, β < N, N N. 2 Discrete multiple orthogonal polynomials For multiple orthogonal polynomials one needs r 2 measures µ 1,..., µ r on R. The polynomials are indexed by a multi-index n = n 1,..., n r N r, with length n = n 1 +... + n r. A type II multiple orthogonal polynomial is defined as a polynomial P n of degree n so that P n xx k dµ 1 x = 0, k = 0, 1,..., n 1 1. P n xx k dµ r x = 0, k = 0, 1,..., n r 1.

This gives a linear system of n homogeneous equations for the n +1 unknown coefficients of P n. The index n is said to be normal if P n is unique up to a multiplicative factor and has exactly degree n. In that case we will always consider monic polynomials. In this paper we will only consider discrete measures on a subset of N. The orthogonality conditions are then more conveniently written as P n j j k w 1 j = 0, k = 0, 1,..., n 1 1, j=0. P n j j k w r j = 0, k = 0, 1,..., n r 1. j=0 Systems of measure µ 1,..., µ r for which all the multi-indices are normal are known as perfect systems. A useful sufficient condition was given by Nikishin. Let w j : [0, [0, be continuous functions. The system w 1, w 2,..., w r is an algebraic Chebyshev system AT system on [0, if w 1 x, xw 1 x,..., x n1 1 w 1 x,... w r x, xw r x,..., x nr 1 w r x is a Chebyshev system on [0, for each multi-index n with n < N. This means that r Q nj 1xw j x j=1 has at most n 1 zeros on [0, for any choice of polynomials Q nj 1 of degree at most n j 1. In an AT system all the n with n < N are normal. All the weights that we use in this paper are AT systems, hence the corresponding monic multiple orthogonal polynomials are unique. In [4] we introduced five families of discrete multiple orthogonal polynomials on a linear lattice. For each of these families we gave a Rodrigues formula, and for r = 2 we gave an explicit expression and a four term recurrence relation, which gives a linear relationship between four multiple orthogonal polynomials with multi-indices n + 1, m, n, m, n, m 1, n 1, m 1. The multiple orthogonal polynomials in each of these five families also satisfy a difference equation of order r + 1. In this paper we will show this for the multiple Charlier and Meixner polynomials for r = 2 and we give the difference equation explicitly. The general case will be considered elsewhere. This difference equation gives a linear relationship for the polynomial of a given degree evaluated at x, x + 1, x + 2 and x + 3. This means that these multiple orthogonal polynomials satisfy both a recurrence relation where the degree changes and a difference equation where the variable changes. This makes them interesting from the viewpoint of bispectral problems.

2.1 Multiple Charlier polynomials For multiple Charlier polynomials C n a x we use r Poisson measures with different parameters a 1,..., a r : µ j = + a k j k! δ k, a j > 0, j = 1,..., r, with a i a j whenever i j. In [4] we showed that there are r raising operators a i w i x w i xc a n x = C a n+ e i x, i = 1,..., r, 1 where w i x = a x i /Γx + 1. An explicit expression for these polynomials for r = 2 was given in [4]: a 1 n1 a 2 n2 C a1,a2 n x n 1 n 2 = n 1 k n 2 l x k+l l=0 We will now show that there is also a lowering operator. 1 a 1 k k! 1 a 2 l Theorem 2.1. For multiple Charlier polynomials with r = 2 we have l!. 2 Cn a1,a2 x = n 1 C a1,a2 n 1 x + n 2 C a1,a2 n 1 x. 3 Proof. Since we are working in an AT system, the monic multiple orthogonal polynomials are unique. We will first show that Cn a1,a2 x = n 1 + n 2 C a1,a2 n x + n 1 1a 1 a 2 C a1,a2 n 1 1 x. 4 If we use summation by parts, then one sees that Cn a1,a2 x is orthogonal to x k for 0 k n 1 2 with respect to w 1, and orthogonal to x k for 0 k n 2 2 with respect to w 2. Both C a1,a2 n 1 x and Ca1,a2 n 1 1 x have the same orthogonality conditions, and they are linearly independent. These n 1 + n 2 2 orthogonality relations show that Cn a1,a2 x = AC a1,a2 n 1 x + BCa1,a2 n x, 1 1 with coefficients A and B that remain to be determined. Comparing the coefficient of x n1+n 2 1 of both sides of this equation shows that A = n 1 + n 2. Comparing coefficients of x n1+n 2 2, for which we use 2 together with x k = k x k 1, gives n 1 a 1 + n 2 a 2 n 1 + n 2 1 = An 1 a 1 + n 2 1a 2 + B,

so that B = n 1 a 1 a 2. This shows that 4 indeed holds. Interchanging the role of a 1 and a 2 and the role of n 1 and n 2 observe that Cn a1,a2 x = Cn a2,a1 2,n 1 x always holds also gives Cn a1,a2 x = n 1 + n 2 C a1,a2 n 1 x n 2 a 1 a 2 C a1,a2 n 1 1 x. 5 Elimination of C a1,a2 n 1 1 x from 4 5 then gives the required relation 3. A combination of the raising operators 1 and the lowering operator 3 gives the difference equation. Theorem 2.2. The multiple Charlier polynomials with r = 2 satisfy the third order difference equation where a 1 a 2 1 2 C a1,a2 n x = [a 1 n 2 1 + a 2 n 1 2 ] C a1,a2 n x, 6 i = Γx + 1 a x i ax i Γx + 1. Proof. First of all, observe that 1 2 = 2 1, so that 1 and 2 are commuting operators. Apply the product 1 2 to 3, then 1 2 C a1,a2 n x = n 1 2 1 C a1,a2 n 1 x + n 2 1 2 C a1,a2 n 1 x. Now use the raising operation 1, then the required result 6 follows. 2.2 Multiple Meixner polynomials of the first kind Meixner polynomials have two parameters β > 0 and 0 < c < 1. We can define two kinds of multiple Meixner polynomials by keeping one parameter fixed and by changing the other parameter [4]. If we keep β fixed and use the measures + β k c k i µ i = δ k, i = 1,..., r, k! where c i c j whenever i j, then we get multiple Meixner polynomials of the first kind M β; c x. In [4] we showed the existence of r raising operators n c i 1 c i w i x; β 1 w i x; βm β; c n x = M β; c n+ e i x, i = 1,..., r, 7 where w i x; β = Γβ + xcx i Γx + 1.

An explicit expression for these polynomials for r = 2 was given in [4]: n1 n2 c1 1 c2 1 M β;c1,c2 x c 1 c 2 n 1+n 2 = β n1+n 2 j=0 j n n 1 k n 2 j k β j k j k c 1 1 c 2 1 c 1 c 2 x j. 8 k! j k! We will now also give a lowering operator for these polynomials. Theorem 2.3. Multiple Meixner polynomials of the first kind for r = 2 have the lowering operator M β;c1,c2 n Proof. We first show that M β;c1,c2 n x = n 1 M β+1;c1,c2 n 1 x = n 1 + n 2 M β+1;c1,c2 x n 1 + n 1 c1 1 c 1 c 2 1 c 2 x + n 2 M β+1;c1,c2 n 1 x, 9 β + n 1 + n 2 1M β+1;c1,c2 n 1 1 x. 10 Summation by parts shows that Mn β;c1,c2 x is orthogonal to x k for 0 k n 1 2 with respect to w 1 x; β + 1, and orthogonal to x k for 0 k n 2 2 with respect to w 2 x; β + 1. Both polynomials M β+1;c1,c2 n x 1 and M β+1;c1,c2 n 1 1 x have the same orthogonality properties and are linearly independent. Hence we can Mn β;c1,c2 x write as a linear combination of these two polynomials M β;c1,c2 n x = AM β+1;c1,c2 n 1 c + BM β+1;c1,c2 n 1 1 x. The n 1 + n 2 2 orthogonality conditions and the two parameters A, B then completely determine the n 1 +n 2 coefficients of the polynomial Mn β;c1,c2 x. Comparing coefficients of x n1+n 2 1 on both sides of the equation gives A = n 1 + n 2. Comparing coefficients of x n1+n 2 2, which can be done by using 8, gives c 1 c 2 n 1 + n 2 1n 1 + n 2 β + n 1 + n 2 1 1 c 1 1 c 2 c 1 c 2 = n 1 + n 2 n 1 + n 2 1 β + n 1 + n 2 1 + B, 1 c 1 1 c 2 which gives B = n 1 [c 1 /1 c 1 c 2 /1 c 2 ]β + n 1 + n 2 1, and thus 10. Changing the role of c 1 and c 2 and the role of n 1 and n 2 also gives M β;c1,c2 n x = n 1 + n 2 M β+1;c1,c2 n 1 x c1 n 2 c 2 1 c 1 1 c 2 β + n 1 + n 2 1M β+1;c1,c2 n 1 1 x. 11

Eliminating M β+1;c1,c2 n 1 1 x from 10 11 then gives the required equation 9. A combination of the raising operators and the lowering operator then gives the difference equation. Theorem 2.4. Multiple Meixner polynomials of the first kind for r = 2 satisfy the third order difference equation [ ] a 1 a 2 β 1 β+1 x = a 2 n 1 β 2 + a 1n 2 β 1 Mn β;c1,c2 x, 12 2 Mn β;c1,c2 where a i = c i /1 c i and Proof. Observe first of all that β i = Γx + 1 Γβ + x 1c x i Apply β 1 β+1 2 to 9 then gives β 1 β+1 2 Mn β;c1,c2 x = n 1 β 2 β+1 β 1 β+1 2 = β 2 β+1 1. 1 M β+1;c1,c2 n 1 cx i Γβ + x Γx + 1. x+n 2 β 1 β+1 2 M β+1;c1,c2 n 1 Now use the raising operations 7, then the required difference equation follows. 2.3 Multiple Meixner polynomials of the second kind Another way to get multiple Meixner polynomials is to keep the parameter c fixed and to change the parameter β. Multiple Meixner polynomials of the β;c second kind M x are related to the measures n µ i = + β i k c k δ k, i = 1,..., r, k! where β i β j Z for every i j. This condition ensures that these measures form a perfect system, so that all multi-indices are normal. The raising operators are given by [4] c 1 cwx; β i 1 β;c wx; β i M n x β e = M i;c n+ e i x, i = 1,..., r, where wx; β = Γβ + xcx Γx + 1. x. 13

An explicit expression was given in [4] for r = 2: n1+n c 1 2 1 Mn β1,β2;c c β 2 n2 β 1 x n1 n 1+n 2 j n 1 k n 2 j k β 1 + n 1 j k = k!j k!β 2 j k j=0 c 1 j c β 1 j x j. 14 The lowering operator for these multiple orthogonal polynomials is given by Theorem 2.5. For multiple Meixner polynomials of the second kind, with r = 2, one has β 1 β 2 Mn β1,β2;c x = n 1 β 1 β 2 n 2 M β1+1,β2;c n 1 x n 2 β 2 β 1 n 1 M β1,β2+1;c n 1 x 15 Proof. If we use summation by parts on the orthogonality relations, then Mn β1,β2;c x is seen to be orthogonal to x k for 0 k n 1 2 with respect to wx; β 1 +1 and orthogonal to x k for 0 k n 2 2 with respect to wx; β 2 + 1. The polynomials M β1+1,β2;c n 1 x and M β1,β2+1;c n 1 x share the same orthogonality conditions. These orthogonality conditions already give n 1 +n 2 2 linear conditions for the n 1 +n 2 coefficients of Mn β1,β2;c x, hence we can write M β1,β2;c n x = AM β1+1,β2;c x + BM β1,β2+1;c x, 16 n 1 n 1 and then we need to determine A and B. x n1+n 2 1 on both sides of 16 gives Comparing the coefficients of A + B = n 1 + n 2, and comparing the coefficients of x n1+n 2 2, for which we use 14, gives n 1 + n 2 1[n 2 β 2 + n 2 1 + n 1 β 1 + n 1 + n 2 1] Solving for A and B gives = A [n 2 β 2 + n 2 1 + n 1 1β 1 + n 1 + n 2 1] + B [n 2 1β 2 + n 2 1 + n 1 β 1 + n 1 + n 2 2]. A = n 1β 1 β 2 n 2 β 1 β 2, B = n 2β 2 β 1 n 1 β 1 β 2. A combination of the raising operators and the lowering operator then gives the third order difference equation.

Theorem 2.6. Multiple Meixner polynomials of the second kind for r = 2 satisfy the third order difference equation β 1 β 2 1 2 Mn β1,β2,c x = 1 c [n 1 β 1 β 2 n 2 2 n 2 β 2 β 1 n 1 1 ] Mn β1,β2,c c x, 17 where Γx + 1 Γβ i + x + 1 i = Γβ i + xc x cx. Γx + 1 Proof. First observe that 1 2 = 2 1, so that both difference operators are commuting. Apply 1 2 to 15, then β 1 β 2 1 2 Mn β1,β2,c x = n 1 β 1 β 2 n 2 2 1 M β1+1,β2;c n 1 x n 2 β 2 β 1 n 1 1 2 M β1,β2+1,c n x. 1 Now apply the raising operations 13, then one gets 17. References [1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1964; Dover, New York, 1972. [2] A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99 1998, 423 447. [3] A. I. Aptekarev, A. Branquinho, W. Van Assche, Multiple orthogonal polynomials for classical weights, manuscript [4] J. Arvesú, J. Coussement, W. Van Assche, Some discrete multiple orthogonal polynomials, J. Comput. Appl. Math. to appear [5] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [6] A. Erdélyi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, 1953. [7] R. Koekoek, R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Reports of the faculty of Technical Mathematics and Informatics no. 98-17, Delft, 1998 math.ca/9602214 at arxiv.org. [8] A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, Berlin, 1991. [9] E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs, vol. 92, Amer. Math. Soc., Providence, RI, 1991.