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
∞ 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
∞ 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
∞ 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 P∞ tn a pair of exponential generating functions g(t) = n=0 gn n! P∞ 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),
P∞ n P∞ 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 c0 r0 0 0 0 ··· !c 1! (c + r ) r ··· 1 1 1! 0 1 0 0 0 2!c 2! (c + r ) 2! (c + 2r ) r 0 ··· P = 2 1! 1 2 2! 0 1 0 . 3! 3! 3! 3!c3 1! (c2 + r3) 2! (c1 + 2r2) 3! (c0 + 3r1) r0 ··· ...... 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 = |Pn+ − xIn+ | p0,1p1,2···pn,n+1 1 1 1 = |xIn+ − Pn+ |. 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) |Pn+1 − xIn+1| = |xIn+1 − Pn+1|.
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
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. Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
t Example 2. The exponential Riordan array A = [1, 1−t ] has general term
( n! n−1 k! k−1 , if n ≥ k ≥ 0, an,k = 0, otherwise,
and the generating functions for its c-sequence and r-sequence are c(t) = 0 and r(t) = (1 + t)2 respectively. t Let (an(x))n≥0 be the associated sequence for f (t) = 1−t . Then a0(x) = 1, a1(x) = x, and for n ≥ 1
an+1(x) = (x − 2n)an(x) − n(n − 1)an−1(x). tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Pn n−k n! n−1 k Example 2(continued). For n ≥ 1, an(x) = k=1(−1) k! k−1 x .
x 1 0 0 0 ··· 0 0
0 2 − x 1 0 0 ··· 0 0
0 2 4 − x 1 0 ··· 0 0 n an+1(x) = (−1) . 0 0 6 6 − x 1 ··· 0 0
......
0 0 0 0 0 ··· n(n − 1) 2n − x
x 1 0
2 − x 2 3 For n = 2, a3(x) = (−1) 0 2 1 = 6x − 6x + x . tu-logo
0 2 4 − x Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 3. Considering the exponential Riordan array A = [(cosh t)y, tanh t], h y q i −1 2 2 1+t we have A = (1 − t ) , ln 1−t . From Lemma 1 we obtain r(t) = 1 − t2 and c(t) = yt. It follows at once that the general entry of production matrix of P i(y − i + 1), if j = i − 1, 0, if j = i, pi,j = 1, if j = i + 1, 0, otherwise.
tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 3(continued). The first rows of P are 0 1 0 0 0 0 ··· y 0 1 0 0 0 ··· 0 2(y − 1) 0 1 0 0 ··· P = . 0 0 3(y − 2) 0 1 0 ··· 0 0 0 4(y − 3) 0 1 ··· ...... ......
y Let (an(x))n≥0 be the Sheffer sequence for ((cosh t) , tanh t). Then a0(x) = 1, a1(x) = x, and for n ≥ 1
tu-logo an+1(x) = xan(x) − n(y − n + 1)an−1(x). Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 3(continued). The determinant formula for an(x) is
−x 1 0 ··· 0 0
y −x 1 ··· 0 0
0 2(y − 1) −x ··· 0 0
n 0 0 3(y − 2) ··· 0 0 (−1) ......
0 0 0 · · · −x 1
0 0 0 ··· (n − 1)(y − n + 2) −x
Note that in this example an(x) are the Cayley continuants of order n. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 4. Considering the exponential Riordan array A = 2 √ [1 − t, t − t ]. Its inverse is A−1 = [ √ 1 , 1 − 1 − 2t ], which 2 1−2t corresponds to the Bessel matrix of the first kind
1 0 0 0 0 ··· 1 1 0 0 0 ··· 3 3 1 0 0 ··· A = . 15 15 6 1 0 ··· 105 105 45 10 1 ··· ...... ......
tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 4(continued). The producttion matrix of A begins −1 1 0 0 0 ··· −1 −2 1 0 0 ··· −3 −3 −3 1 0 ··· P = −15 −12 −6 −4 1 ··· , −105 −75 −30 −10 −5 ··· ...... ...... whose general entry is (n+1)! 2n−2k, if 0 ≤ k ≤ n, k!(n−k+1)2n−k n−k pn,k = 1, if k = n + 1, 0, otherwise. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
4. Examples
Example 4(continued). t2 Let (an(x))n≥0 be the Sheffer sequence for (1−t, t− 2 ). Then a0(x) = 1, and for n ≥ 1, an(x) = |xIn − Pn|, where Pn is the n-th principal submatrix of the production matrix P. For n = 3, we have
x + 1 −1 0
2 3 a3(x) = |xI3 − P3| = 1 x + 2 −1 = 15 + 15x + 6x + x .
3 3 x + 3
Note that in this example an(x) are the Bessel polynomials
with exponents in decreasing order. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples
Thank you for your attention
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