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., [1–7]). 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., [8–12]).

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=0∞gntn and f(t)=∑n=0∞fntn are formal power series with g0=1, f0=0, and f1≠0. 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=0∞bntn. 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,k≥0 can be characterized by two sequences A=(ai)i≥0 and Z=(zi)i≥0 such that, for n,k≥0(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,j≥0 (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,n≠0. Then R is a Riordan array if and only if its production matrix P is of the form
(7)P=(z0a00000⋯z1a1a0000⋯z2a2a1a000⋯z3a3a2a1a00⋯z4a4a3a2a1a0⋯⋮⋮⋮⋮⋮⋮⋱),
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))n≥0 be a sequence of polynomials where rn(x) is of degree n and rn(x)=∑k=0nrn,kxk. We say that (rn(x))n≥0 is a polynomial sequence of Riordan type if the coefficient matrix (rn,k)n,k≥0 is an element of the Riordan group; that is, there exists a Riordan array (g(t),f(t)) such that (rn,k)n,k≥0=(g(t),f(t)). In this case, we say that (rn(x))n≥0 is the polynomial sequence associated with the Riordan array (g(t),f(t)).

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

Lemma 3.
Let (rn(x))n≥0 be the polynomial sequence associated with a Riordan array (g(t),f(t)), and let r(t,x)=∑n=0∞rn(x)tn be its generating function. Then
(9)r(t,x)=g(t)1-xf(t).

In [15], Luzón introduced a new notation T(f∣g) 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 [18]. In [19], 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)i≥0 and the A-sequence (ai)i≥0. Let (pn(x))n≥0 be the polynomial sequence associated with R-1. Then (pn(x))n≥0 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 n≥1, pn(x) is given by the following Hessenberg determinant:
(11)pn(x)=(-1)na0-n ×|z0-xa0000⋯00z1a1-xa000⋯00z2a2a1-xa00⋯00z3a3a2a1-xa0⋯00z4a4a3a2a1-x⋯00⋮⋮⋮⋮⋮⋱⋮⋮zn-2an-2an-3an-4an-5⋯a1-xa0zn-1an-1an-2an-3an-4⋯a2a1-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)(z0a0000⋯z1a1a000⋯z2a2a1a00⋯z3a3a2a1a0⋯⋮⋮⋮⋮⋮⋱)(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⋮)+(a0000⋯a1a000⋯a2a1a00⋯a3a2a1a0⋯⋮⋮⋮⋮⋱) ×(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)⋮)=(0000⋯x000⋯0x00⋯00x0⋯⋮⋮⋮⋮⋱)(p1(x)p2(x)p3(x)p4(x)⋮).
The previous matrix equation can be rewritten as
(15)(a0000⋯a1-xa000⋯a2a1-xa00⋯a3a2a1-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|a00000⋯0(x-z0)p0(x)a1-xa0000⋯0-z1p0(x)a2a1-xa000⋯0-z2p0(x)a3a2a1-xa00⋯0-z3p0(x)a4a3a2a1-xa0⋯0-z4p0(x)⋮⋮⋮⋮⋮⋱⋮⋮an-2an-3an-4an-5an-6⋯a0-zn-2p0(x)an-1an-2an-3an-4an-5⋯a1-x-zn-1p0(x)|=p0(x)a0-n|a00000⋯0x-z0a1-xa0000⋯0-z1a2a1-xa000⋯0-z2a3a2a1-xa00⋯0-z3a4a3a2a1-xa0⋯0-z4⋮⋮⋮⋮⋮⋱⋮⋮an-2an-3an-4an-5an-6⋯a0-zn-2an-1an-2an-3an-4an-5⋯a1-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))n≥0 be the polynomial sequence associated with R-1=(g(t),f(t))-1. Then p0(x)=1, and for all n≥1,
(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))n≥0 be the polynomial sequence associated with 𝒜=(1/(1+rt2),at/(1+rt2)). Then (pn(x))n≥0 satisfies the recurrence relation:
(21)pn(x)=axpn-1(x)-rpn-2(x), n≥2,
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|-ax10000⋯0r-ax1000⋯00r-ax100⋯000r-ax10⋯0000r-ax1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-ax|.
If a=1, r=-1, then pn(x) becomes the Fibonacci polynomials:
(23)Fn(x)=(-1)n|-x10000⋯0-1-x1000⋯00-1-x100⋯000-1-x10⋯0000-1-x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-x|.
If a=2, r=-1, then pn(x) gives the Pell polynomials:
(24)Pn(x)=Fn(2x)=(-1)n|-2x10000⋯0-1-2x1000⋯00-1-2x100⋯000-1-2x10⋯0000-1-2x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-2x|.
In case a=2, r=1, pn(x) becomes the Chebyshev polynomials of the second kind:
(25)Un(x)=(-1)n|-2x10000⋯01-2x1000⋯001-2x100⋯0001-2x10⋯00001-2x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-2x|.

Example 7.
Considering the Riordan array ℬ=((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 ℬ-1 are
(26)A(t)=1+rt2a, Z(t)=(b+r)ta.

Let (pn(x))n≥0 be the polynomial sequence associated with ℬ=((1-bt2)/(1+rt2),at/(1+rt2)). Then (pn(x))n≥0 satisfies the recurrence relation:
(27)pn(x)=axpn-1(x)-rpn-2(x), n≥3,
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|-ax10000⋯0b+r-ax1000⋯00r-ax100⋯000r-ax10⋯0000r-ax1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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|-2x10000⋯02-2x1000⋯001-2x100⋯0001-2x10⋯00001-2x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-2x|.

In case a=3, b=0, r=2, pn(x) give the Fermat polynomials (see [15]):
(30)fn(x)=(-1)n|-3x10000⋯02-3x1000⋯002-3x100⋯0002-3x10⋯00002-3x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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))n≥0 be the polynomial sequence associated with 𝒞=((1-bt)/(1+rt2),at/(1+rt2)). Then (pn(x))n≥0 satisfies the recurrence relation:
(32)pn(x)=axpn-1(x)-rpn-2(x), n≥3,
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-ax10000⋯0r-ax1000⋯00r-ax100⋯000r-ax10⋯0000r-ax1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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|1-2x10000⋯01-2x1000⋯001-2x100⋯0001-2x10⋯00001-2x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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|-1-2x10000⋯01-2x1000⋯001-2x100⋯0001-2x10⋯00001-2x1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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))n≥0 be the polynomial sequence associated with 𝒟=((1-bt-ct2)/(1+t2),at/(1+t2)). Then (pn(x))n≥0 satisfies the recurrence relation:
(37)pn(x)=axpn-1(x)-pn-2(x), n≥3,

with initial condition p0(x)=1, and p1(x)=ax-b, p2(x)=axp1(x)-(c+1)p0(x)=a2x2-abx-c-1. For n≥1, we have
(38)pn(x)=(-1)n|b-ax10000⋯0c+1-ax1000⋯001-ax100⋯0001-ax10⋯00001-ax1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-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-ax10000⋯0c+1-ax1000⋯001-ax100⋯0001-ax10⋯00001-ax1⋯0⋮⋮⋮⋮⋮⋮⋱⋮000000⋯1000000⋯-ax|, n≥1,
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.