JDM Journal of Discrete Mathematics 2090-9845 2090-9837 Hindawi Publishing Corporation 734836 10.1155/2013/734836 734836 Research Article Determinant Representations of Polynomial Sequences of Riordan Type Yang Sheng-liang Zheng Sai-nan Cheon Gi Sang Department of Applied Mathematics Lanzhou University of Technology Lanzhou 730050 China lut.cn 2013 20 03 2013 2013 12 11 2012 19 01 2013 27 01 2013 2013 Copyright © 2013 Sheng-liang Yang and Sai-nan Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

1. Introduction

The concept of a Riordan array is very useful in combinatorics. The infinite triangles of Pascal, Catalan, Motzkin, and Schröder are important and meaningful examples of Riordan array, and many others have been proposed and developed (see, e.g., ). In the recent literature, Riordan arrays have attracted the attention of various authors from many points of view and many examples and generalizations can be found (see, e.g., ).

A Riordan array denoted by (g(t),f(t)) is an infinite lower triangular matrix such that its column k  (k=0,1,2,) has generating function g(t)f(t)k, where g(t)=n=0gntn and f(t)=n=0fntn are formal power series with g0=1, f0=0, and f10. That is, the general term of matrix R=(g(t),f(t)) is rn,k=[tn]g(t)f(t)k; here [tn]h(t) denotes the coefficient of tn in power series h(t). Given a Riordan array (g(t),f(t)) and column vector B=(b0,b1,b2,)T, the product of (g(t),f(t)) and B gives a column vector whose generating function is g(t)  b(f(t)), where b(t)=n=0bntn. If we identify a vector with its ordinary generating function, the composition rule can be rewritten as (1)(g(t),f(t))b(t)=g(t)b(f(t)). This property is called the fundamental theorem for Riordan arrays and this leads to the matrix multiplication for Riordan arrays: (2)(g(t),f(t))(h(t),l(t))=(g(t)h(f(t)),l(f(t))). The set of all Riordan arrays forms a group under the previos operation of a matrix multiplication. The identity element of the group is (1,t). The inverse element of (g(t),f(t)) is (3)(g(t),f(t))-1=(1g(f-(t)),f-(t)), where f-(t) is compositional inverse of f(t).

A Riordan array R=(g(t),f(t))=(rn,k)n,k0 can be characterized by two sequences A=(ai)i0 and Z=(zi)i0 such that, for n,k0(4)rn+1,0=z0rn,0+z1rn,1+z2rn,2+,rn+1,k+1=a0rn,k+a1rn,k+1+a2rn,k+2+. If A(t) and Z(t) are the generating functions for the A- and Z-sequences, respectively, then it follows that [9, 13] (5)g(t)=11-tZ(f(t)),f(t)=tA(f(t)). If the inverse of R=(g(t),f(t)) is R-1=(d(t),h(t)), then the A- and Z-sequences of R are (6)A(t)=th(t),Z(t)=1h(t)(1-d(t)).

For an invertible lower triangular matrix R, its production matrix (also called its Stieltjes matrix; see [11, 14]) is the matrix P=R-1R¯, where R¯ is the matrix R with its first row removed. The production matrix P can be characterized by the matrix equality RP=DR, where D=(δi+1,j)i,j0 (δ is the usual Kronecker delta).

Lemma 1 (see [<xref ref-type="bibr" rid="B5">14</xref>]).

Assume that R=(rn,k) is an infinite lower triangular matrix with rn,n0. Then R is a Riordan array if and only if its production matrix P is of the form (7)P=(z0a00000z1a1a0000z2a2a1a000z3a3a2a1a00z4a4a3a2a1a0), where (a0,a1,a2,) is the A-sequence and (z0,z1,z2,) is the Z-sequence of the Riordan array R.

Definition 2.

Let (rn(x))n0 be a sequence of polynomials where rn(x) is of degree n and rn(x)=k=0nrn,kxk. We say that (rn(x))n0 is a polynomial sequence of Riordan type if the coefficient matrix (rn,k)n,k0 is an element of the Riordan group; that is, there exists a Riordan array (g(t),f(t)) such that (rn,k)n,k0=(g(t),f(t)). In this case, we say that (rn(x))n0 is the polynomial sequence associated with the Riordan array (g(t),f(t)).

Letting rn(x)=k=0nrn,kxk, n0, then in matrix form we have (8)(r0,0000r1,0r1,100r2,0r2,1r2,20r3,0r3,1r3,2r3,3)(1xx2x3)=(r0(x)r1(x)r2(x)r3(x)). Hence, by using (1), we have the following lemma.

Lemma 3.

Let (rn(x))n0 be the polynomial sequence associated with a Riordan array (g(t),f(t)), and let r(t,x)=n=0rn(x)tn be its generating function. Then (9)r(t,x)=g(t)1-xf(t).

In , Luzón introduced a new notation T(fg) to represent the Riordan arrays and gave a recurrence relation for the family of polynomials associated to Riordan arrays. In recent works [16, 17], a new definition by means of a determinant form for Appell polynomials is given. Sequences of Appell polynomials are special of the Sheffer sequences . In , the author obtains a determinant representation for the Sheffer sequence. The aim of this work is to propose a similar approach for polynomial sequences of Riordan type, which are special of the generalized Sheffer sequences [12, 18]. A determinant representation for polynomial sequences of Riordan type is obtained by using production matrix of Riordan array. In fact, we will show that the general formula for the polynomial sequences of Riordan type can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, determinant expressions for some classical polynomial sequences such as Fibonacci, Pell, and Chebyshev are derived, and a unified determinant expression for the four kinds of Chebyshev polynomials [20, 21] is established.

2. Main Theorem

In this section we are going to develop our main theorem.

Theorem 4.

Let R=(g(t),f(t)) be a Riordan array with the Z-sequence (zi)i0 and the A-sequence (ai)i0. Let (pn(x))n0 be the polynomial sequence associated with R-1. Then (pn(x))n0 satisfies the recurrence relation: (10)a0pn(x)=xpn-1(x)-a1pn-1(x)-a2pn-2(x)--an-1p1(x)-zn-1p0(x),n>1, with initial condition p0(x)=1, and a0p1(x)=x-z0. In general, for all n1, pn(x) is given by the following Hessenberg determinant: (11)pn(x)=(-1)na0-n×|z0-xa000000z1a1-xa00000z2a2a1-xa0000z3a3a2a1-xa000z4a4a3a2a1-x00zn-2an-2an-3an-4an-5a1-xa0zn-1an-1an-2an-3an-4a2a1-x|.

Proof.

Let R=(g(t),f(t)) and R-1=(g(t),f(t))-1=(d(t),h(t)). Then from definition and (3), we have g(0)=1 and d(t)=1/g(f-(t)). Hence d(0)=1 and p0(x)=1. Letting E=(1,x,x2,)T, then R-1E=(p0(x),p1(x),p2(x),)T and DE=(x,x2,x3,)T, where D=(δi+1,j). Letting P be the production matrix of R, then RP=DR, and PR-1=R-1D. Thus PR-1E=R-1DE, and PR-1E=xR-1E. In matrix form, we have (12)(z0a0000z1a1a000z2a2a1a00z3a3a2a1a0)(p0(x)p1(x)p2(x)p3(x))=(xp0(x)xp1(x)xp2(x)xp3(x)).

Using the block matrix method, we get (13)p0(x)(z0z1z2z3)+(a0000a1a000a2a1a00a3a2a1a0)×(p1(x)p2(x)p3(x)p4(x))=(xp0(x)xp1(x)xp2(x)xp3(x)).

Since (14)(xp0(x)xp1(x)xp2(x)xp3(x))=(xp0(x)000)+(0xp1(x)xp2(x)xp3(x)),(0xp1(x)xp2(x)xp3(x))=(0000x0000x0000x0)(p1(x)p2(x)p3(x)p4(x)). The previous matrix equation can be rewritten as (15)(a0000a1-xa000a2a1-xa00a3a2a1-xa0)(p1(x)p2(x)p3(x)p4(x))=p0(x)(x-z0-z1-z2-z3). Therefore, a0p1(x)=x-z0, and for n>1, we have (16)an-1p1(x)++a2pn-2(x)+(a1-x)pn-1(x)+a0pn(x)=-zn-1p0(x), or equivalently (17)a0pn(x)=(x-a1)pn-1(x)-a2pn-2(x)--an-1p1(x)-zn-1p0(x).

By applying the Cramer’s rule, we can work out the unknown pn(x) operating with the first n equations in (15): (18)pn(x)=a0-n|a000000(x-z0)p0(x)a1-xa00000-z1p0(x)a2a1-xa0000-z2p0(x)a3a2a1-xa000-z3p0(x)a4a3a2a1-xa00-z4p0(x)an-2an-3an-4an-5an-6a0-zn-2p0(x)an-1an-2an-3an-4an-5a1-x-zn-1p0(x)|=p0(x)a0-n|a000000x-z0a1-xa00000-z1a2a1-xa0000-z2a3a2a1-xa000-z3a4a3a2a1-xa00-z4an-2an-3an-4an-5an-6a0-zn-2an-1an-2an-3an-4an-5a1-x-zn-1|.

After transferring the last column to the first position, an operation which introduces the factor (-1)n-1, the theorem follows.

Corollary 5.

Let R=(g(t),f(t)) be a Riordan array with production matrix P. Let (pn(x))n0 be the polynomial sequence associated with R-1=(g(t),f(t))-1. Then p0(x)=1, and for all n1, (19)pn(x)=a0-ndet(xIn-Pn), where a0=p0,1, Pn is the principal submatrix of order n of the production matrix P and In is the identity matrix of order n.

3. Applications

A useful application of Theorem 4 is to find the determinant expression of a well-known sequence. We illustrate the ideal in the following examples. In the final paragraph, we will give a unified determinant expression for the four kinds of Chebyshev polynomials.

Example 6.

Considering the Riordan array 𝒜=(1/(1+rt2),at/(1+rt2)), we have (1/(1+rt2),at/(1+rt2))(1/(1-xt))=1/(1-axt+rt2). The generating functions of the A- and Z-sequences of 𝒜-1 are (20)A(t)=1+rt2a,Z(t)=rta.

Let (pn(x))n0 be the polynomial sequence associated with 𝒜=(1/(1+rt2),at/(1+rt2)). Then (pn(x))n0 satisfies the recurrence relation: (21)pn(x)=axpn-1(x)-rpn-2(x),n2, with initial condition p0(x)=1, and p1(x)=ax. In general, pn(x) is also given by the following Hessenberg determinant: (22)pn(x)=(-1)n|-ax100000r-ax100000r-ax100000r-ax100000r-ax100000001000000-ax|. If a=1, r=-1, then pn(x) becomes the Fibonacci polynomials: (23)Fn(x)=(-1)n|-x100000-1-x100000-1-x100000-1-x100000-1-x100000001000000-x|. If a=2, r=-1, then pn(x) gives the Pell polynomials: (24)Pn(x)=Fn(2x)=(-1)n|-2x100000-1-2x100000-1-2x100000-1-2x100000-1-2x100000001000000-2x|. In case a=2, r=1, pn(x) becomes the Chebyshev polynomials of the second kind: (25)Un(x)=(-1)n|-2x1000001-2x1000001-2x1000001-2x1000001-2x100000001000000-2x|.

Example 7.

Considering the Riordan array =((1-bt2)/(1+rt2),at/(1+rt2)), we have ((1-bt2)/(1+rt2),at/(1+rt2))(1/(1-xt))=(1-bt2)/(1-axt+rt2). Then the generating functions of the A- and Z-sequences of -1 are (26)A(t)=1+rt2a,Z(t)=(b+r)ta.

Let (pn(x))n0 be the polynomial sequence associated with =((1-bt2)/(1+rt2),at/(1+rt2)). Then (pn(x))n0 satisfies the recurrence relation: (27)pn(x)=axpn-1(x)-rpn-2(x),  n3, with initial condition p0(x)=1, and p1(x)=ax, p2(x)=a2x2-b-r.

In general, pn(x) is also given by the following Hessenberg determinant: (28)pn(x)=(-1)n|-ax100000b+r-ax100000r-ax100000r-ax100000r-ax100000001000000-ax|.

If a=2, b=r=1, then pn(x) become the Chebyshev polynomials of the first kind 2Tn(x)-0n: (29)2Tn(x)=(-1)n|-2x1000002-2x1000001-2x1000001-2x1000001-2x100000001000000-2x|.

In case a=3, b=0, r=2, pn(x) give the Fermat polynomials (see ): (30)fn(x)=(-1)n|-3x1000002-3x1000002-3x1000002-3x1000002-3x100000001000000-3x|.

Example 8.

Considering the Riordan array 𝒞=((1-bt)/(1+rt2),at/(1+rt2)), we have ((1-bt)/(1+rt2),at/(1+rt2))(1/(1-xt))=(1-bt)/(1-axt+rt2). The generating functions of the A- and Z-sequences of 𝒞-1 are (31)A(t)=1+rt2a,Z(t)=b+rta.

Let (pn(x))n0 be the polynomial sequence associated with 𝒞=((1-bt)/(1+rt2),at/(1+rt2)). Then (pn(x))n0 satisfies the recurrence relation: (32)pn(x)=axpn-1(x)-rpn-2(x),n3, with initial condition p0(x)=1, and p1(x)=ax-b, p2(x)=a2x2+abx-r.

In general, pn(x) is also given by the following Hessenberg determinant: (33)pn(x)=(-1)n|b-ax100000r-ax100000r-ax100000r-ax100000r-ax100000001000000-ax|.

If a=2, b=1, r=1, then pn(x) becomes the Chebyshev polynomials of the third kind Vn(x): (34)Vn(x)=(-1)n|1-2x1000001-2x1000001-2x1000001-2x1000001-2x100000001000000-2x|.

If a=2, b=-1, r=1, then pn(x) gives the Chebyshev polynomials of the fourth kind Wn(x): (35)Wn(x)=(-1)n|-1-2x1000001-2x1000001-2x1000001-2x1000001-2x100000001000000-2x|.

Finally, considering the Riordan array 𝒟=((1-bt-ct2)/(1+t2),at/(1+t2)), we have ((1-bt-ct2)/(1+t2),at/(1+t2))(1/(1-xt))=(1-bt-ct2)/(1-axt+t2). Then the generating functions of the A- and Z-sequences of 𝒟-1 are (36)A(t)=1+t2a,Z(t)=b+(c+1)ta.

Let (pn(x))n0 be the polynomial sequence associated with 𝒟=((1-bt-ct2)/(1+t2),at/(1+t2)). Then (pn(x))n0 satisfies the recurrence relation: (37)pn(x)=axpn-1(x)-pn-2(x),n3,

with initial condition p0(x)=1, and p1(x)=ax-b, p2(x)=axp1(x)-(c+1)p0(x)=a2x2-abx-c-1. For n1, we have (38)pn(x)=(-1)n|b-ax100000c+1-ax1000001-ax1000001-ax1000001-ax100000001000000-ax|.

Therefore we can give, now, the following.

Definition 9.

The Chebyshev polynomial of degree n, denoted by Cn(x,a,b,c), is defined by (39)C0(x,a,b,c)=1,Cn(x,a,b,c)=(-1)n|b-ax100000c+1-ax1000001-ax1000001-ax1000001-ax100000001000000-ax|,n1, where Cn(x,a,b,c) is represented by a Hessenberg determinant of order n.

Note that  Cn(x,2,0,1)=2Tn(x), Cn(x,2,0,0)=Un(x), Cn(x,2,1,0)=Vn(x), and Cn(x,2,-1,0)=Wn(x). Hence, Definition 9 can be considered as a unified form for the four kinds of Chebyshev polynomials.

Acknowledgments

The authors wish to thank the editor and referee for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).

Cheon G.-S. Kim H. Shapiro L. W. Combinatorics of Riordan arrays with identical A and Z sequences Discrete Mathematics 2012 312 12-13 2040 2049 10.1016/j.disc.2012.03.023 MR2920864 ZBL1243.05007 He T. X. Parametric Catalan numbers and Catalan triangles Linear Algebra and its Applications 2013 438 3 1467 1484 10.1016/j.laa.2012.10.001 MR2997825 He T. X. Hsu L. C. Shiue P. J. S. The Sheffer group and the Riordan group Discrete Applied Mathematics 2007 155 15 1895 1909 10.1016/j.dam.2007.04.006 MR2351975 ZBL1123.05007 Shapiro L. W. Getu S. Woan W. J. Woodson L. C. The Riordan group Discrete Applied Mathematics 1991 34 1–3 229 239 10.1016/0166-218X(91)90088-E MR1137996 ZBL0754.05010 Shapiro L. W. Bijections and the Riordan group Theoretical Computer Science 2003 307 2 403 413 10.1016/S0304-3975(03)00227-5 MR2022586 ZBL1048.05008 Sprugnoli R. Riordan arrays and the Abel-Gould identity Discrete Mathematics 1995 142 1–3 213 233 10.1016/0012-365X(93)E0220-X MR1341448 ZBL0832.05007 Sprugnoli R. Riordan arrays and combinatorial sums Discrete Mathematics 1994 132 1–3 267 290 10.1016/0012-365X(92)00570-H MR1297386 ZBL0814.05003 He T. X. Sprugnoli R. Sequence characterization of Riordan arrays Discrete Mathematics 2009 309 12 3962 3974 10.1016/j.disc.2008.11.021 MR2537389 ZBL1228.05014 Merlini D. Rogers D. G. Sprugnoli R. Verri M. C. On some alternative characterizations of Riordan arrays Canadian Journal of Mathematics 1997 49 2 301 320 10.4153/CJM-1997-015-x MR1447493 ZBL0886.05013 Munarini E. Riordan matrices and sums of harmonic numbers Applicable Analysis and Discrete Mathematics 2011 5 2 176 200 10.2298/AADM110609014M MR2867317 Peart P. Woan W. J. Generating functions via Hankel and Stieltjes matrices Journal of Integer Sequences 2000 3 2 article 00.2.1 MR1778992 ZBL0961.15018 Wang W. Wang T. Generalized Riordan arrays Discrete Mathematics 2008 308 24 6466 6500 10.1016/j.disc.2007.12.037 MR2466952 ZBL1158.05008 Deutsch E. Ferrari L. Rinaldi S. Production matrices and Riordan arrays Annals of Combinatorics 2009 13 1 65 85 10.1007/s00026-009-0013-1 MR2529720 ZBL1229.05015 Deutsch E. Ferrari L. Rinaldi S. Production matrices Advances in Applied Mathematics 2005 34 1 101 122 10.1016/j.aam.2004.05.002 MR2102277 ZBL1064.05012 Luzón A. Morón M. A. Recurrence relations for polynomial sequences via Riordan matrices Linear Algebra and its Applications 2010 433 7 1422 1446 10.1016/j.laa.2010.05.021 MR2680268 ZBL1250.11029 Costabile F. A. Longo E. A determinantal approach to Appell polynomials Journal of Computational and Applied Mathematics 2010 234 5 1528 1542 10.1016/j.cam.2010.02.033 MR2610369 ZBL1200.33020 Yang Y. Determinant representations of Appell polynomial sequences Operators and Matrices 2008 2 4 517 524 10.7153/oam-02-32 MR2468879 ZBL1166.15003 Roman S. The Umbral Calculus 1984 111 New York, NY, USA Academic Press Pure and Applied Mathematics MR741185 Yang S. l. Recurrence relations for the Sheffer sequences Linear Algebra and its Applications 2012 437 12 2986 2996 10.1016/j.laa.2012.07.015 MR2966613 ZBL1251.05015 Aghigh K. Masjed-Jamei M. Dehghan M. A survey on third and fourth kind of Chebyshev polynomials and their applications Applied Mathematics and Computation 2008 199 1 2 12 10.1016/j.amc.2007.09.018 MR2415797 ZBL1134.33300 Comtet L. Advanced Combinatorics 1974 Dordrecht, The Netherlands D. Reidel Publishing MR0460128