Sheffer sequences Exponential Riordan array Recurrent relations Examples Recurrence relations for the Sheffer sequences Sheng-liang Yang Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, P.R. China E-mail address: [email protected] 2012 Shanghai Conference on Algebraic Combinatorics August 17-22, 2012 Shanghai Jiao Tong University tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples Outline 1 Sheffer sequences 2 Exponential Riordan array 3 Recurrent relations 4 Examples tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples Introduction In this talk, using the production matrix of an exponential Ri- ordan array [g(t); f (t)], we give a recurrence relation for the Shef- fer sequence for the ordered pair (g(t); f (t)) . We also develop a new determinant representation for the general term of the Sheffer sequence. As applications, determinant expressions for some classical Sheffer polynomial sequences are derived. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 1. Sheffer sequences polynomial sequence: p0(x), p1(x), p2(x), ··· , pk(x), ··· . deg pk(x) = k, for all k ≥ 0. Sheffer sequence: s0(x), s1(x), s2(x), ··· , sk(x), ··· . Let g(t) be an invertible series and let f (t) be a delta series; we say that the polynomial sequence sn(x) is the Sheffer sequence for the pair (g(t); f (t)) if and only if 1 n X t 1 xf (t) sn(x) = e : (1) n! g(f (t)) n=0 where f (t) is the compositional inverse of f (t). tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 1. Sheffer sequences Appell sequence: a0(x), a1(x), a2(x), ··· , an(x), ··· . The Sheffer sequence for (g(t); t) is the Appell sequence for g(t). If an(x) is Appell for g(t), then 1 X tn 1 a (x) = ext: n n! g(t) n=0 sequence of binomial type ( associated sequence ): b0(x), b1(x), ··· . The Sheffer sequence for (1; f (t)) is the associ- ated sequence for f (t). If bn(x) is associated to f (t), then 1 n X t xf (t) bn(x) = e : n! tu-logo n=0 Sheffer sequences Exponential Riordan array Recurrent relations Examples 2. Exponential Riordan array Exponential Riordan array [g(t); f (t)] An exponential Riordan array D = (dn;k)n;k≥0 is defined by P1 tn a pair of exponential generating functions g(t) = n=0 gn n! P1 tn and f (t) = n=0 fn n! with g0 = f1 = 1 and f0 = 0, such that tn d = g(t)f (t)k: n;k n! The exponential Riordan group is the set of all exponential Riordan array with the operation being matrix multiplication. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 2. Exponential Riordan array Production matrix For an invertible lower triangular matrix R, its production ma- trix (also called its Stieltjes matrix) is the matrix P = R−1R; where R is the matrix R with its first row removed. the c-sequence (cn)n≥0 and the r-sequence (rn)n≥0 of expo- nential Riordan array A = [g(t); f (t)] are determined by r(f (t)) = f 0(t); c(f (t)) = g0(t)=g(t); P1 n P1 n where c(t) = n=0 cnt and r(t) = n=0 rnt . tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 2. Exponential Riordan array Lemma 1 Let A = [g(t); f (t)] be an exponential Riordan array and let the c- sequence and the r-sequence of exponential Riordan array A are (cn)n≥0 and (rn)n≥0, respectively. Then the the production matrix P = (pi;j)i;j≥0 of A is determined by ( where defining c−1 = 0) i! p = (c + jr ); i;j j! i−j i−j+1 0 1 c0 r0 0 0 0 ··· B !c 1! (c + r ) r ··· C B 1 1 1! 0 1 0 0 0 C B C B 2!c 2! (c + r ) 2! (c + 2r ) r 0 ··· C P = B 2 1! 1 2 2! 0 1 0 C : B 3! 3! 3! C B 3!c3 1! (c2 + r3) 2! (c1 + 2r2) 3! (c0 + 3r1) r0 ··· C @ . A tu-logo . .. Sheffer sequences Exponential Riordan array Recurrent relations Examples 2. Exponential Riordan array Lemma 2 Pn k If sn(x) = k=0 sn;kx is Sheffer for (g(t); f (t)), then the coeffi- cients sn;k are the elements of the exponential Riordan array [ 1 ; f (t)]. If xn = Pn b s (x), then b are the elements g(f (t)) k=0 n;k k n;k of the exponential Riordan array [g(t); f (t)]. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 3 Let B be an invertible lower triangular matrix with production ma- −1 Pn k trix P = (pn;k)n;k≥0. Let B = A = (an;k) and an(x) = k=0 an;kx . Then (an(x))n≥0 satisfies the recurrence relation of the form: pn;n+1an+1(x) = −(pn;n − x)an(x) − pn;n−1an−1(x) − · · · −pn;1a1(x) − pn;0a0(x); with initial condition a0(x) = a0;0 and p0;1a1(x) = x − p0;0. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 3 ( continued) For n ≥ 0, an+1(x) is also given by the following determinant formula p − x p ··· 0 0;0 0;1 p0;1 p1;1 − x ··· 0 (−1)n+1 an+1(x) = . p0;1p1;2···pn;n+1 . .. pn;0 pn;1 ··· pn;n − x (−1)n+1 = jPn+ − xIn+ j p0;1p1;2···pn;n+1 1 1 1 = jxIn+ − Pn+ j: p0;1p1;2···pn;n+1 1 1 tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 4 Let [g(t); f (t)] be an exponential Riordan array with the c- sequences (ci)i≥0 and r-sequences (ri)i≥0. Let (an(x))n≥0 be the Sheffer polynomial sequence for (g(t); f (t)). Then (an(x))n≥0 sat- isfies the recurrence relation of the form: n! an+1(x) = (x − c0 − nr1)an(x) − (n−1)! (c1 + (n − 1)r2)an−1(x) n! n! − · · · − 2! (cn−2 + 2rn−1)a2(x) − 1! (cn−1 + rn)a1(x) − n!cna0(x); with initial condition a0(x) = 1 and a1(x) = x − c0. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 4 ( continued) For n ≥ 0, an+1(x) is given by the following determinant formula c0 − x 1 ··· 0 1! 1!c1 1! (c0 + r1) − x ··· 0 n+1 an+1(x) = (−1) . .. n!c n! (c + r ) ··· n! (c + nr ) − x n 1! n−1 n n! 0 1 n+1 = (−1) jPn+1 − xIn+1j = jxIn+1 − Pn+1j: tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 5 Let the r-sequence of [1; f (t)] be (ri)i≥0, and let (an(x))n≥0 be the associated polynomial sequence to f (t). Then (an(x))n≥0 satis- fies the recurrence relation of the form: n! n! a (x) = (x − nr )a (x) − r a (x) − · · · − r a (x); n+1 1 n (n − 2)! 2 n−1 0! n 1 with initial condition a0(x) = 1, and a1(x) = x. For n ≥ 0, x −1 0 ··· 0 0 x − 1! r −1 ··· 0 0! 1 2! 2! 0 − r2 x − r1 ··· 0 an+1(x) = 0! 1! . .. n! n! n! tu-logo 0 − 0! rn − 1! rn−1 ··· x − (n−1)! r1 Sheffer sequences Exponential Riordan array Recurrent relations Examples 3. Recurrent relations Theorem 6 Let the c-sequence of [g(t); t] be (ci)i≥0, and let (an(x))n≥0 be the Appell polynomial sequence for g(t). Then (an(x))n≥0 satisfies n!c1 n!cn−1 an+1(x) = (x − c0)an(x) − (n−1)! an−1(x) − · · · − 1! a1(x) − n!cna0(x); with initial condition a0(x) = 1, and a1(x) = x − c0. For n ≥ 0, x − c0 −1 0 ··· 0 −1!c1 x − c0 −1 ··· 0 −2!c − 2! c x − c ··· 0 an+1(x) = 2 1! 1 0 . .. n! n! −n!cn − 1! cn−1 − 2! cn−2 ··· x − c0 tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 4. Examples v t2 Example 1. The exponential Riordan array A = [e 2 ; t] has general term 8 n−k ( v ) 2 < n! 2 k! n−k ; if n ≥ k and n − k even; an;k = ( 2 )! :0; otherwise; and the generating functions for its c-sequence and r-sequence are c(t) = vt and r(t) = 1 respectively. vt2 Let (an(x))n≥0 be the Appell sequence for g(t) = e 2 . Then a0(x) = 1, a1(x) = x, and for n ≥ 1 an+1(x) = xan(x) − nvan−1(x); tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 4. Examples Example 1(continued). with the general formula −x 1 0 0 0 ··· 0 0 v −x 1 0 0 ··· 0 0 0 2v −x 1 0 ··· 0 0 n+1 0 0 3v −x 1 ··· 0 0 an+1(x) = (−1) : . .. 0 0 0 0 0 · · · −x 1 0 0 0 0 0 ··· nv −x −x 1 0 3 3 For n = 2, we have a3(x) = (−1) v −x 1 = −3vx + x : 0 2v −x When v = 2, we have an(2x) = Hn(x), which are the Hermite tu-logo polynomials.