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Moments of Orthogonal Polynomials and Exponential Generating MOMENTS OF ORTHOGONAL POLYNOMIALS AND EXPONENTIAL GENERATING FUNCTIONS IRA M. GESSEL AND JIANG ZENG Dedicated to the memory of Richard Askey Abstract. Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of (q = 1) classical orthogonal polynomials, and study those cases in which the expo- nential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. 1. Introduction Many of the most important sequences in enumerative combinatorics—the factorials, derangement numbers, Bell numbers, Stirling polynomials, secant numbers, tangent numbers, Eulerian polynomials, Bernoulli numbers, and Catalan numbers—arise as moments of well-known orthogonal polynomials. With the exception of the Bernoulli and Catalan numbers, these orthogonal polynomials are all Sheffer type, see [39, 42]. One characteristic of these sequences is that their ordinary generating functions have simple continued fractions. For some recent work on the moments of classical orthog- onal polynomials we refer the reader to [9, 14, 32, 10, 11, 23]. There is another sequence which appears in a number of enumerative applications, and which also has a simple continued fraction. The Genocchi numbers may be defined by arXiv:2107.00255v2 [math.CA] 9 Jul 2021 ∞ xn ∞ x2n+2 x G = G = x tan . n n! 2n+2 (2n + 2)! 2 n=0 n=0 X X So Gn = 0 when n is odd or n = 0, G2 = 1, G4 = 1, G6 = 3, G8 = 17, G10 = 155, and G12 = 2073. A closely related sequence is the median Genocchi numbers H2n+1, which first ap- peared in Seidel’s work [36]. These numbers do not seem to have a simple exponential generating function and may be defined by H = 1, and for n 1, 1 ≥ ⌊(n+1)/2⌋ n H = ( 1)k−1 G . (1) 2n+1 − 2k 1 2n+2−2k k=1 X 1 − MOMENTS OF ORTHOGONAL POLYNOMIALS 2 So H1 = 1, H3 = 1, H5 = 2, H7 = 8, H7 = 56, H11 = 608, H13 = 9440; see [38, 16, 18, 20]. In his combinatorial approach to orthogonal polynomials [39, p.V-10], Viennot briefly alludes to the monic orthogonal polynomials whose moments are the Genocchi numbers (i.e., the nth moment is G2n+2) and the median Genocchi numbers H2n+1, respectively, but he does not give any explicit formula for them. Many years ago, one of the authors (IG) was learning about continued fractions and their connection to orthogonal polynomials, and saw in Viennot’s work [39] the simple continued fraction for the Genocchi numbers. He wondered what the corresponding orthogonal polynomials were and wrote to Richard Askey, asking if he knew. Askey immediately wrote back to say that they were a special case of the continuous dual Hahn polynomials, and this paper grew out of an attempt to understand his reply. A comprehensive discussion of the combinatorial properties of Genocchi numbers has been given by Viennot [38]. In particular, he showed that the Genocchi numbers and median Genocchi numbers H2n+1 have the S-fraction expansions ∞ G t2n = S(t2;12, 1 2, 22, 2 3, 32, 3 4, 42,...), (2) 2n+2 · · · n=0 X∞ 2n 2 2 2 2 2 2 2 H2n+1 t = S(t ; 1 , 1 , 2 , 2 , 3 , 3 ,...). (3) n=0 X They are somewhat more recondite than the other sequences mentioned above, and unlike the closely related tangent numbers, there is no straightforward derivation of any of their combinatorial interpretations directly from the exponential generating function, and in fact it is nontrivial to prove that they are integers. Some recent papers (see [4, 21, 29]) have shown there are renewed interest on Genocchi numbers and median Genocchi numbers. As the Wilson polynomials are the most general (q = 1) classical orthogonal poly- nomials, we shall first consider the moments of the Wilson polynomials. In the general case, there are four parameters a,b,c,d and the ordinary the generating function of these moments can be expressed as a hypergeometric series. We show that these mo- ments have a simple exponential generating function when when a =0 or a =1/2. We then consider the continuous dual Hahn polynomials and the (continuous) Hahn polynomials along with their rescaled versions. In particular, we show that the mo- ments of a rescaled version of the continuous dual Hahn polynomials are the generalized Dumont-Foata polynomials, which are a refinement of both Genocchi numbers and me- dian Genocchi numbers. It would be possible to derive the moment generating function for Hahn polynomials from that for Wilson polynomials, but we will give a separate derivation and use our method for variety. MOMENTS OF ORTHOGONAL POLYNOMIALS 3 2. Wilson polynomials The monic Wilson polynomials Wn(x) (see [2, 40, 25]) are defined by (a + b) (a + c) (a + d) W (x2)=( 1)n n n n W (x2; a,b,c,d) (4) n − (a + b + c + d + n 1) − n where f n, n + a + b + c + d 1, a + ix, a ix W (x2; a,b,c,d)= F ;1 . 4 3 − a + b, a + c,− a + d − If a,b,c,df are positive or a = ¯b and/or c = d¯ and the real parts are positive, then the orthogonality reads as follows: 1 ∞ Γ(a + ix)Γ(b + ix)Γ(c + ix)Γ(d + ix) 2 W (x2)W (x2) dx 2π Γ(2ix) m n Z0 n!Γ(n + a + b) Γ(n + c + d) = ··· δ . (5a) (n + a + b + c + d 1) Γ(2n + a + b + c + d) mn − n Consider the corresponding moment sequence wn(a) := wn(a,b,c,d) of the Wilson polynomials defined by 1 ∞ Γ(a + ix)Γ(b + ix)Γ(c + ix)Γ(d + ix) 2 x2ndx 2π Γ(2ix) Z0 Γ(a + b)Γ(a + c)Γ(a + d )Γ(b + c)Γ(b + d)Γ(c + d) = w (a) . (5b) n Γ(a + b + c + d) It follows that w (a)=1 and for n 0, 0 ≥ (a + b)(a + c)(a + d) w (a)= w (a + 1) a2w (a). (6) n+1 a + b + c + d n − n Here is the formal approach to the Wilson polynomials. We define a linear functional 2n L on even polynomials by L (x )= wn(a). Lemma 1. For k 0, ≥ 2n (a + b)k (a + c)k (a + d)k L (x (a + ix)k (a ix)k)= wn(a + k). (7) − (a + b + c + d)k Proof. We prove this by induction on k. It is clear for k = 0. For k = 1 we derive from (6) that (a + b)(a + c)(a + d) L (x2n(a2 + x2)) = a2w (a)+ w (a)= w (a + 1). n n+1 a + b + c + d n MOMENTS OF ORTHOGONAL POLYNOMIALS 4 Suppose this is true for k. Then L (x2n(a + ix) (a ix) ) k+1 − k+1 = (a + k)2 + x2) L (x2n(a + ix) (a ix) k − k (a + b)k (a + c)k (a + d)k 2 = (a + k) wn(a + k)+ wn+1(a + k) . (a + b + c + d)k The result for k + 1 follows then from (6). Lemma 2. For integers m, n 0, ≥ L ((a + ix) (a ix) (b + ix) (b ix) ) m − m n − n (a + b) (a + c) (a + d) (b + c) (b + d) = m+n m m n n . (a + b + c + d)m+n Proof. This can be proved directly from the case m = n = 0 of (5a): replacing a with a + m and b with b + n in 1 ∞ Γ(a + ix)Γ(b + ix)Γ(c + ix)Γ(d + ix) 2 dx 2π Γ(2ix) Z0 Γ(a + b)Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)Γ(c + d) = . (8) Γ(a + b + c + d) together with the definition of L and (8) gives Lemma 2. Theorem 3. We have (a + b) (a + c) (a + d) (b + c) (b + d) (c + d) L (W (x2)W (x2)) = n! n n n n n n δ . n m (a + b + c + d + n 1) (a + b + c + d) mn − n 2n Proof. We evaluate L W˜ (x2; a,b,c,d)(b + ix) (b ix) n m − m (a + b)m(b + c)m(b + d)m n, n + a + b + c + d 1, a + b + m = 3F2 − − ;1 (a + b + c + d) a + b, a + b + c + d + m m (a + b) (b + c) (b + d) (1 n c d) ( m) = m m m − − − n − n (a + b + c + d) (a + b) (1 n a b c d m) m n − − − − − − n (by the Pfaff-Saalsch¨utz theorem) (a + b) (b + c) (b + d) (c + d) ( m) = m m m n − n . (a + b + c + d)m+n(a + b)n This is 0 for m < n, which implies orthogonality. MOMENTS OF ORTHOGONAL POLYNOMIALS 5 By Favard’s theorem [8] there are complex numbers bn and λn such that xWn(x)= Wn+1(x)+ bnWn(x)+ λnWn−1(x). (9) Corollary 4. We have λ = A C , b = A + C a2, n n−1 n n n n − where (n + a + b + c + d 1)(n + a + b)(n + a + c)(n + a + d) A = − , n (2n + a + b + c + d 1)(2n + a + b + c + d) − n(n + b + c 1)(n + b + d 1)(n + c + d 1) C = − − − n (2n + a + b + c + d 2)(2n + a + b + c + d 1). − − Proof. From Theorem 3 we derive that 2 2 2 2 2 L (x Wn−1(x )Wn(x )) L (Wn(x ) ) λn = 2 2 = 2 2 = An−1Cn.
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