We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation

Vol. 19 (2018), No. 2, pp. 1019–1033 DOI: 10.18514/MMN.2018.2682

DETERMINANTS CONTAINING POWERS OF POLYNOMIAL SEQUENCES

H.-H. LEUNG Received 25 September, 2018

Abstract. We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas polynomials and certain orthogonal polynomials.

These identities naturally generalize the determinant identities obtained by Alfred, Carlitz, Pro- dinger, Tangboonduangjit and Thanatipanonda.

2010Mathematics Subject Classification: 11B39; 11C20

Keywords: Fibonacci polynomial, Lucas polynomial, Chebyshev polynomial, determinant iden- tity

1. INTRODUCTION

Let .Fn/n0 be the Fibonacci sequence. Let m1 and n be any nonnegative integer. LetŒF_n^m_CiCj0i;jmbe the.mC1/.mC1/matrix with entriesFnCiCj, 0i; j m. In 1966, Carlitz [4] derived the following determinant identity for the matrixŒF_n^m_CiCj0i;jm:

det.ŒF_n^m_CiCj0i;jm/D. 1/^.nC1/.^mC₂¹/.F₁^mF₂^{m 1} Fm/²

m

Y

iD0

m i

!

: (1.1) This identity is related to the problems posted by Alfred [1, p. 48] in 1963 and Parker [9, p. 303] in 1964 respectively. Letsandkbe any integers. Tangboonduangjit and Thanatipanonda [12] generalized the determinant identity (1.1) as follows:

det.ŒF_s^m_{Ck.nCiCj /}_0i;jm/D. 1/^{.sCk nC1/}.^mC₂¹/.F_k^mF_2k^{m 1} F_mk/²

m

Y

iD0

m i

! : (1.2) LetFn^hmibe therising powersof the Fibonacci numbers defined by

F_n^hmiWDFnFnC1 FnCm 1:

The author was supported in part by the UAEU Startup Grant 2016, Grant No. G00002235.

c 2018 Miskolc University Press

Prodinger [10] obtained the following determinant identity for the matrix ŒF_n^hm_Cⁱ_iCj0i;jm:

det.ŒF_n^hm_Cⁱ_iCj0i;jm/D. 1/ⁿ.^mC₂¹/C.^mC₃²/.F1F2 Fm/^mC1: (1.3) Tangboonduangjit and Thanatipanonda [11] generalized the determinant identity (1.3) as follows:

det.ŒF_n^hm_Cⁱ_iCj_{0i;jd 1}/D. 1/ⁿ.^d2/C.^dC3¹/

d 1

Y

iD1

.FiFmC1 i/^{d i}

2.d 1/

Y

iDd 1

F_n^hm_C^C_i ^{1 di}: (1.4) where d 2. It is worthwhile to note that Tangboonduangjit and Thanatipanonda [11,12] derived the determinant identities more generally, for matrices whose entries include (rising) powers of terms that satisfy a second-order linear recurrence relation with constant coefficients. By using analogous techniques in determinant calculus, we derive determinant identities for matrices whose entries are (rising) powers of polynomials that satisfy certain recurrence relations. As corollaries, we provide de- terminant identities for matrices whose entries are (rising) powers of Fibonacci poly- nomials, Lucas polynomials and certain orthogonal polynomials. As an application, we obtain new identities in the case of Fibonacci numbers. For example, forn1, by Corollary2, we get

det

h 1 FnCiCj

i

0i;jm

D. 1/ⁿ.^mC2¹/Qm

iD0F_i^{2.m i /}_C1 Q

0i;jmFnCiCj

: (1.5)

2. MAIN RESULTS

Definition 1. Letp; q; r; a; bandc be any real numbers. LetZ0D f0; 1; 2; : : :g. The sequence of polynomials in variablex,

P.x/.p; q; rIa; b; c/WD.Pn/n2Z; is defined by

P0WDp; P1WDqxCr; PnC2WD.axCb/PnC1CcPn;forn2Z0: Forn < 0,Pnis defined by

PnWD axCb

c PnC1C1 cPnC2: The discriminant_P is defined by

P WD.q² apq/x²C.2qr apr bpq/xC.r² bpr cp²/:

Theorem 1. LetP.x/.p1; q1; r1Ia; b; c/D.Pn/n2Z, Q.x/.p2; q2; r2Ia; b; c/D .Qn/n2ZandU.x/.0; 0; 1Ia; b; c/D.Un/n2Zbe the sequences of polynomials defined by real numbersp1; p2; q1; q2; r1; r2; a; b; cwherec¤0. Then

PsCiQsCj PsQsCiCj D. c/^s.P1Qj P0QjC1/Ui

for all integerss; i; j.

Proof. We prove it by induction oni. It is trivial foriD0. IfiD1, we have PsC1 QsCjC1

Ps QsCj

D

axCb c

1 0

Ps QsCj

Ps 1 QsCj 1

D D

axCb c

1 0

s

P1 QjC1

P0 Qj

; (2.1)

PsC1 QsCjC1

Ps QsCj

D

axCb c

1 0

1

PsC2 QsCjC2

PsC1 QsCjC1

D D

axCb c

1 0

s

P1 QjC1

P0 Qj

(2.2) fors0ands < 0respectively. We take the determinants on both sides of (2.1) and (2.2) to get

PsC1QsCj PsQsCjC1D. c/^s.P1Qj P0QjC1/

D. c/^s.P1Qj P0QjC1/U1: (2.3) Fori > 1, we assume that the identity is true fori 1andi 2. We have

PsCiQsCj PsQsCiCj

D ˇ ˇ ˇ ˇ

PsCi QsCiCj

Ps QsCj

ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ

.axCb/PsCi 1CcPsCi 2 .axCb/QsCiCj 1CcQsCiCj 2

Ps QsCj

ˇ ˇ ˇ ˇ

D.axCb/

ˇ ˇ ˇ ˇ

PsCi 1 Q_{sCjC.i 1/}

P_s Q_sCj ˇ ˇ ˇ ˇCc

ˇ ˇ ˇ ˇ

PsCi 2 Q_{sCjC.i 2/}

P_s Q_sCj ˇ ˇ ˇ ˇ

D.axCb/. c/^s.P₁Q_j P₀Q_jC1/U_{i 1}Cc. c/^s.P₁Q_j P₀Q_jC1/U_{i 2} D. c/^s.P1Qj P0QjC1/..axCb/Ui 1CcUi 2/

D. c/^s.P1Qj P0QjC1/Ui:

Fori < 0, we assume that the identity is true foriC1andiC2. We have PsCiQsCj PsQsCiCj D

ˇ ˇ ˇ ˇ

PsCi QsCiCj

Ps QsCj

ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ

axCb

c PsCiC1C¹_cPsCiC2 axCb

c QsCiCjC1C¹_cQsCiCjC2

Ps QsCj

ˇ ˇ ˇ ˇ

D axCb c

ˇ ˇ ˇ ˇ

PsCiC1 QsCiCjC1

Ps QsCj

ˇ ˇ ˇ ˇC1

c ˇ ˇ ˇ ˇ

PsCiC2 QsCiCjC2

Ps QsCj

ˇ ˇ ˇ ˇ

D axCb

c . c/^s.P1Qj P0QjC1/UiC1C1

c. c/^s.P1Qj P0QjC1/UiC2

D. c/^s.P1Qj P0QjC1/ axCb

c UiC1C1 cUiC2

D. c/^s.P1Qj P0QjC1/Ui:

Remark1. We recover thegeneralized Catalan Identityby Melham and Shannon [8] (see also Tangboonduangjit and Thanatipanonda [12, Proposition 1]) by substi- tutingxD1in Theorem1.

Corollary 1. Let P.x/.p; q; rIa; b; c/D.Pn/n2Z and U.x/.0; 0; 1Ia; b; c/D .Un/n2Z be the sequences of polynomials defined by real numbers p; q; r; a; b; c wherec¤0. Then

PjP1 P0PjC1D_PUj; (2.4) PsCiPsCj PsPsCiCj D. c/^s_PUiUj (2.5) where_P is the discriminant ofP.x/.p; q; rIa; b; c/.

Proof. By settingsD0; j D1; p1Dp2Dp; q1Dq2Dq; r1Dr2Drin Theorem 1, we get

PiP1 P0PiC1D.P1P1 P0P2/Ui: (2.6) We note thatP0Dp,P1DqxCr andP2D.axCb/P1CcP0D.axCb/.qxC r/Ccp. Hence, we obtain (2.4) by simplifying (2.6). On the other hand, by setting p1Dp2Dp; q1Dq2Dq; r1Dr2Drin Theorem1, we get

PsCiPsCj PsPsCiCj D. c/^s.P1Pj P0PjC1/Ui D. c/^s_PUiUj

in which the last equality is based on (2.4).

Lemma 1. Letm1. LetBi; Di be polynomials in variablex,Ai; Ci be rational functions in variablex, for0im. LetŒ.AjBiCCjDi/^m0i;jmbe the.mC 1/.mC1/matrix with entries.AjBiCCjDi/^m; 0i; j m. Then we have the following determinant identity:

det.Œ.AjBiCCjDi/^m0i;jm/D D Y

0i <jm

.BiDj BjDi/.AiCj AjCi/

m

Y

iD0

m i

!

: (2.7)

Proof. We invoke the following result by Krattenthaler [7, Lemma 10] (see also Tangboonduangjit and Thanatipanonda [12, Lemma 3]:

det.Œ.cjdiC1/^m0i;jm/D Y

0i <jm

.di dj/.ci cj/

m

Y

iD0

m i

!

(2.8)

wherecj; di are real numbers for0i; j m. First, we prove the lemma for poly- nomialsAi; Bi; Ci; Di for all 0i m. For the values ofxsuch that Cj ¤0and Di ¤0for0i; j m, let

cj DAj

Cj

; diD Bi

Di

;for0i; j m:

We note that

det.Œ.cjdiC1/^m0i;jm/Ddet

hAjBiCCjDi

CjDi

mi

0i;jm

D Y

0im

1 .CiDi/^m

det.Œ.AjBiCCjDi/^m0i;jm/: (2.9)

Also, we have Y

0i <jm

.di dj/.ci cj/

D Y

0i <jm

BiDj BjDi

DiDj

AiCj AjCi

CiCj

D Y

0im

1 .CiDi/^m

Y

0i <jm

.BiDj BjDi/.AiCj AjCi/

: (2.10)

By (2.8), (2.9), (2.10), we get (2.7) as desired.

Based on the facts that there are only a finite number of roots for Cj; Di where 0i; j mand the determinant of a matrix with polynomial entries is a continuous function inx, the equality (2.7) still holds true for the values ofx such thatCj D0 orDiD0for someiorj.

Next, we assume thatAi andCi are rational functions for all0im. We write Ai andCi as follows:

Ai DEi

Fi

; CiD Gi

Hi

for0i m;

whereEi; Fi; Gi; Hi are all polynomials for0i m. For the values ofxsuch that Fi¤0andHi¤0for all0im, we get

det.Œ.AjBiCCjDi/^m0i;jm/Ddet

hHjEjBiCDiFjGj

F_jH_j

mi

0i;jm

D Y

0im

1 .FiHi/^m

det.Œ..HjEj/BiC.GjFj/Di/^m0i;jm/

D Y

0im

1 .FiHi/^m

Y^m

iD0

m i

!

Y

0i <jm

.BiDj BjDi/.HiEiGjFj HjEjGiFi/

DY^m

iD0

m i

!

Y

0i <jm

.BiDj BjDi/ Ei

Fi Gj

Hj

Ej

Fj Gi

Hi

DY^m

iD0

m i

!

Y

0i <jm

.BiDj BjDi/ AiCj AjCi :

For the values ofxsuch thatFiD0orHiD0for somei, the equality still holds true as the determinant of a matrix with polynomial entries is a continuous function.

Theorem 2. Lets; k; nbe any integers,m1. LetP.x/.p; q; rIa; b; c/D.Pn/_n2Z andU.x/.0; 0; 1Ia; b; c/D.Un/n2Zbe the sequences of polynomials defined by real numbers p; q; r; a; b; c where c ¤ 0. The determinant of the matrix ŒP_s^m_{Ck.nCiCj /}0i;jmis given by

det.ŒP_s^m_{Ck.nCiCj /}0i;jm/ D. 1/^{.sCk nC1/}.^mC₂¹/.^mC21/

P c^{.sCk n/}.^mC₂¹/C2k.^mC₃¹/

m

Y

iD0

m i

!

U_k.i^{2.m i /}_C1/

whereP is the discriminant ofP.x/.p; q; rIa; b; c/.

Proof. By substituting sD km; i Dkj⁰; j Ds⁰Ck.nCmCi⁰/ into (2.5) in Corollary1and then replacings⁰; i⁰; j⁰bys; i; j respectively, we get

P_{k.j m/}P_{sCk.nCi /} P _kmP_{sCk.nCiCj /}D. c/ ^km_PU_kjU_{sCk.nCmCi /}; P_{sCk.nCiCj /}DP_{k.j m/}

P _km P_{sCk.nCi /}C . c/ ^km_PU_kj

P _km U_{sCk.nCmCi /}: (2.11)

By substitutingsDs⁰Ck.nCmCi⁰/; iDk.j⁰ i⁰/; j D kminto Theorem1and then replacings⁰; i⁰; j⁰bys; i; j respectively, we get

P_{sCk.nCi /}U_{sCk.nCmCj /} P_{sCk.nCj /}U_{sCk.nCmCi /} D. c/^{sCk.nCmCi /}.U1P km U0P kmC1/Uk.j i /

D. c/^{sCk.nCmCi /}P kmUk.j i /:

(2.12)

By substitutingsDki⁰; iDk.j⁰ i⁰/; j D kminto Theorem1and then replacing i⁰; j⁰byi; j respectively, we get

P_{k.i m/}U_kj P_{k.j m/}U_kiD. c/^kiP _kmU_{k.j i /}: (2.13) By (2.11), we get

det.ŒP_s^m_{Ck.nCiCj /}0i;jm/ Ddet

hP_{k.j m/}

P km P_{sCk.nCi /}C . c/ ^km_PU_kj

P km U_{sCk.nCmCi /}mi

0i;jm

: (2.14) By (2.7), the term in (2.14) becomes

m

Y

iD0

m i

! Y

0i <jm

.P_{sCk.nCi /}U_{sCk.nCmCj /} P_{sCk.nCj /}U_{sCk.nCmCi /}/ . c/ ^km_P

P²_km Pk.i m/Ukj Pk.j m/Uki

: (2.15) By (2.12), (2.13), the term in (2.15) becomes

m

Y

iD0

m i

! Y

0i <jm

. 1/^{sCk nC1}c^{sCk.nC2i /}_PU_{k.j i /}²

: (2.16)

As a consequence, we get the desired result by standard counting arguments.

Remark2. We recover Theorem 5 in the work of Tangboonduangjit and Thana- tipanonda [12] by substitutingxD1in Theorem2.

Next, we look at other determinant identities.

Lemma 2. Letm1. LetBi; Di be polynomials in variablexfor0im. Let Aj; Cj be rational functions in variablexfori2Z. Let.di/1irand.ei/1irbe sequences of integers. Then

det

h Y^m

fDjC1

A_dfB_iCC_dfD_i

j

Y

gD1

A_egB_iCC_egD_ii

0i;jm

D Y

0i <jm

BiDj BjDi Y

1ijm

CeiA_dj AeiC_dj : Proof. By the factorization method of Krattenthaler [7, Section 4], it is plain to get the following identity:

det

h Y^m

fDjC1

XiCF_f

j

Y

gD1

XiCGgi

0i;jm

D Y

0i <jm

Xj Xi Y

1ijm

Fj Gi

(2.17) where Xi for 0i m, Dj; Ej for 1j m are some indeterminates. For the values ofxsuch thatDi¤0andAj ¤0for0imandj 2Z, let

Xi D Bi

Di

; Fj DCd_j

Ad_j

; Gj DCe_j

Aej

for0imand1j m. By similar reasoning as in the proof of Lemma1, we get the desired result by clearing the denominators on both sides of (2.17). For the values ofx which are the roots ofDi orAj for somei orj, the equality still holds true based on the fact that the determinant of a matrix with polynomial entries is a

continuous function.

Theorem 3. Lets; k; nbe any integers,m1. LetP.x/.p; q; rIa; b; c/D.Pn/_n2Z andU.x/.0; 0; 1Ia; b; c/D.Un/n2Zbe the sequences of polynomials defined by real numbersp; q; r; a; b; c wherec¤0. Let.di/1imand.ei/1imbe sequences of integers. Then

det

h Y^m

fDjC1

P_sCk.nCiCdf/

j

Y

gD1

P_sCk.nCiCeg/i

0i;jm

D. _P/.^mC21/. c/^{.sCk n/}.^mC2¹/Ck.^mC3¹/

m

Y

lD1

U_kl^{mC1 l} Y

1ijm

. c/^kdjU_{k.ei dj/} whereP is the discriminant ofP.x/.p; q; rIa; b; c/.

Proof. By (2.11), Lemma 2 and Corollary 1, the theorem can be proved in the

same way as in the proof the Theorem2.

Lemma 3. Letm1. Let Ai; Bi are polynomials in variablex for0i m.

LetCi; Di be rational functions in variablexfor0im. Then, det

h 1 AiDjCBiCj

i

0i;jm

D

Q

0i <jm.AiBj AjBi/.CiDj DiCj/ Q

0i;jm.AiDjCBiCj/ provided that the denominators on both sides of the identity are nonzero.

Proof. First, we invoke a result of Krattenthaler [7, Theorem 12]. That is, det

h 1 xiCyj

i

0i;jm

D

Q

0i <jm.xi xj/.yi yj/ Q

0i;jm.xiCyj/ (2.18) wherexiandyiare indeterminates for0i; jm. We first assume thatAi; Bi; Ci; Di

are all polynomials for all0im. For the values ofxsuch thatBi; Di are nonzero for all0i m, let

xi DAi

Bi

; yi D Ci

Di

for0im:

By similar reasoning as shown in the proof of Lemma1, we get the desired result by some algebraic simplification for the cases whereAi; Bi; Ci; Di are polynomials for all0i m.

We extend the proof to the cases whereCi andDi are rational functions by the same arguments as in the proof of Lemma1, based on the fact that the determinant of a matrix with rational functions as entries is a continuous function provided that the denominators on both sides of the identity are nonzero.

Theorem 4. Lets; k; nbe any integers,m1. LetP.x/.p; q; rIa; b; c/D.Pn/n2Z

andU.x/.0; 0; 1Ia; b; c/D.Un/_n2Zbe the sequences of polynomials defined by real numbers p; q; r; a; b; c where c ¤ 0. The determinant of the matrix Œ1=P_{sCk.nCiCj /}0i;jmis given by

det

h 1 P_{sCk.nCiCj /}

i

0i;jm

Dc^{.sCk n/}.^mC2¹/C2k.^mC3¹/.^mC21/

P

Qm

iD0U_k.i^{2.m i /}_C1/

. 1/^{.sCk n/}.^mC12 /Q

0i;jmP_{sCk.nCiCj /} where_P is the discriminant ofP.x/.p; q; rIa; b; c/, provided that the denominat- ors on both sides of the identity are nonzero.

Proof. The proof is essentially the same as the proof of Theorem2 by applying (2.11), (2.12), (2.13) to Lemma3and some standard counting arguments.

LetAbe ammmatrix. LetA_k.i; j /be the determinant of thekksubmatrix ofAwhose first entry is at the position of thei^{t h}-row and thej^{t h}-column ofA.

Lemma 4. LetAbe ammmatrix whose entries are rational functions in vari- ablex. Then

Am.1; 1/Am 2.2; 2/DAm 1.1; 1/Am 1.2; 2/ Am 1.2; 1/Am 1.1; 2/:

Proof. We invoke the Desnanot-Jacobi identity [3] for a matrixAof sizemm with indeterminates as entries.

Am.1; 1/Am 2.2; 2/DAm 1.1; 1/Am 1.2; 2/ Am 1.2; 1/Am 1.1; 2/:

To extend this result to the case where the matrixAhas rational functions as entries, we simply use the same strategy as in the proof of Lemma1.

Let m 1. The rising powers of a sequence of polynomials P.x/.p; q; rIa; b; c/D.Pn/n2Zis denoted byP_n^hmi, which is defined by

P_n^hmiWDPnPnC1 PnCm 1:

Theorem 5. Let n be any integer. Let m 1 and d 1. Let P.x/.p; q; rIa; b; c/D.Pn/_n2ZandU.x/.0; 0; 1Ia; b; c/D.Un/_n2Zbe the sequen- ces of polynomials defined by real numbersp; q; r; a; b; cwherec¤0. Then

det.ŒP_n^hm_Cⁱ_iCj_{0i;jd 1}/

D. 1/ⁿ.^d₂/C.^dC₃¹/c^{.nCd 2/}.^d₂/.^d2/

P

d 1

Y

iD1

UiUrC1 id i

2.d 1/

Y

iDd 1

P_n^hm_C^C₁^{1 di}

where_P is the discriminant ofP.x/.p; q; rIa; b; c/.

Proof. The proof is based on induction on d, Lemma 4 and Theorem 1. It is essentially identical to the proof of Theorem 2.1 in the work of Tangboonduangjit

and Thanatipanonda [11] and hence we skip it.

If we setpDqDbD0andrDaDcD1, then we get the sequence of Fibonacci polynomials inP.x/.0; 0; 1I1; 0; 1/D.Fn.x//n2Z where the sequence.Fn.x//n2Z

is defined by

F0.x/0; F1.x/1; FnC2.x/DxFnC1.x/CFn.x/:

We recover the Fibonacci numbers and Pell numbers by evaluatingFn.x/atxD1 andxD2respectively. We note that_PD1andU.x/.0; 0; 1I1; 0; 1/D.Fn.x//n2Z. By Theorem2, Theorem3, Theorem4and Theorem5, we get the following corol- lary:

Corollary 2. Letm1andd1. Lets; k; nbe any integers. Let.di/1imand .ei/1imbe sequences of integers. Then

det.Œ.FsCk.nCiCj /.x//^m0i;jm/D. 1/^{.sCk nC1/}.^mC2¹/

m

Y

iD0

m i

!

.F_.iC1/k.x//^{2.m i /};

det

h Y^m

fDjC1

F_sCk.nCiCdf/.x/

j

Y

gD1

F_sCk.nCiCeg/.x/i

0i;jm

D. 1/^{.sCk nC1/}.^mC₂¹/Ck.^mC₃¹/

m

Y

lD1

.F_kl.x//^{mC1 l} Y

1ijm

. 1/^kdjF_{k.ei dj/}.x/;

det

h 1

FsCk.nCiCj /.x/

i

0i;jm

D. 1/^{.sCk n/}.^mC₂¹/Qm

iD0.F_k.iC1/.x//^{2.m i /} Q

0i;jmFsCk.nCiCj /.x/ ; det.Œ.FnCiCj.x//^hmi_{0i;jd 1}/

D. 1/ⁿ.^d₂/C.^dC₃¹/

d 1

Y

iD1

Fi.x/FmC1 i.x/d i

2.d 1/

Y

iDd 1

.FnC1.x//^{hmC1 di}:

Remark3. We recover the identities (1.2) and (1.4) by setting xD1in the first identity and the last identity in Corollary2respectively.

Remark4. We recover the results shown by Alfred [2] by settingxD1; sD0; kD 1; nD0anddi 0,ei1for all1i; j min the second identity in Corollary2.

Remark5. We get the identity (1.5) by settingxD1,sD0,kDnD1in the third identity in Corollary2.

If we setpD2,qDaDcD1andrDbD0, then we get the sequence of Lucas polynomials inP.x/.2; 1; 0I1; 0; 1/D.Ln.x//n2Zwhere the sequence.Ln.x//n2Z

is defined by

L0.x/2; L1.x/x; LnC2.x/DxLnC1.x/CLn.x/:

We recover the Lucas numbers by evaluatingLn.x/atxD1. We note that_P D . x² 4/andU.x/.0; 0; 1I1; 0; 1/D.Fn.x//n2Z. By Theorem2, Theorem3, The- orem4and Theorem5, we get the following corollary:

Corollary 3. Letm1andd1. Lets; k; nbe any integers. Let.di/1imand .ei/1imbe sequences of integers.

det.Œ.L_{sCk.nCiCj /}.x//^m0i;jm/ D. 1/^{.sCk n/}.^mC2¹/.x²C4/.^mC21/

m

Y

iD0

m i

!

.F_.iC1/k.x//^{2.m i /};

det

h Y^m

fDjC1

L_sCk.nCiCdf/.x/

j

Y

gD1

L_sCk.nCiCeg/.x/i

0i;jm

D. 1/^{.sCk n/}.^mC12 /Ck.^mC13 /.x²C4/.^mC12 /

m

Y

lD1

.F_kl.x//^{mC1 l} Y

1ijm

. 1/^kdjF_{k.ei dj/}.x/;

det

h 1

L_{sCk.nCiCj /}.x/

i

0i;jm

D. 1/^{.sCk nC1/}.^mC₂¹/.x²C4/.^mC21/ Q

0i;jmL_{sCk.nCiCj /}.x/

m

Y

iD0

.F_k.iC1/.x//^{2.m i /} det.Œ.LnCiCj.x//^hmi0i;jd 1/

D. 1/^.nC1/.^d₂/C.^dC₃¹/.x²C4/.^d2/

d 1

Y

iD1

Fi.x/FmC1 i.x/d i

2.d 1/

Y

iDd 1

.LnC1.x//^{hmC1 di}:

If we set pDqD1, aD2, cD 1 andr Db D0, then we get the sequence of Chebyshev polynomials of the first kind inP.x/.1; 1; 0I2; 0; 1/D.Tn.x//n2Z

where the sequence.Tn.x//n2Zis defined by

T0.x/1; T1.x/x; TnC2.x/D2xTnC1.x/ Tn.x/:

We note thatP D. x²C1/.

If we setpD1,aDqD2,cD 1andrDbD0, then we get the sequence of Chebyshev polynomials of the second kind inP.x/.1; 2; 0I2; 0; 1/D.Sn.x//_n2Z where the sequence.Sn.x//n2Z is defined by

S0.x/1; S1.x/2x; SnC2.x/D2xSnC1.x/ Sn.x/:

We note that_P D. 2x²C1/.

We note that

U.x/.0; 0; 1I2; 0; 1/D.Un.x//n2Z

where

Un.x/DSn 1.x/forn2Z:

We get two corollaries by Theorem2, Theorem3, Theorem4and Theorem5.

Corollary 4. Letm1andd1. Lets; k; nbe any integers. Let.di/1imand .ei/1imbe sequences of integers. Then

det.Œ.T_{sCk.nCiCj /}.x//^m0i;jm/D.x² 1/.^mC21/

m

Y

iD0

m i

!

.S_{.iC1/k 1}.x//^{2.m i /};

det

h Y^m

fDjC1

T_sCk.nCiCdf/.x/

j

Y

gD1

T_sCk.nCiCeg/.x/i

0i;jm

D.x² 1/.^mC21/

m

Y

lD1

.S_{kl 1}.x//^{mC1 l} Y

1ijm

S_{k.ei dj/ 1}.x/;

det

h 1

TsCk.nCiCj /.x/

i

0i;jm

D. x²C1/.^mC21/Qm

iD0.S_{k.iC1/ 1}.x//^{2.m i /} Q

0i;jmTsCk.nCiCj /.x/ ; det.Œ.TnCiCj.x//^hmi_{0i;jd 1}/

D. 1/^d.^d₂/C.^dC₃¹/. x²C1/.^d2/

d 1

Y

iD1

Si 1.x/Sm i.x/d i

2.d 1/

Y

iDd 1

.TnC1.x//^{hmC1 di}: Corollary 5. Letm1andd1. Lets; k; nbe any integers. Let.di/1imand .ei/1imbe sequences of integers. Then

det.Œ.S_{sCk.nCiCj /}.x//^m0i;jm/D.2x² 1/.^mC21/

m

Y

iD0

m i

!

.S_{.iC1/k 1}.x//^{2.m i /};

det

h Y^m

fDjC1

S_sCk.nCiCdf/.x/

j

Y

gD1

S_sCk.nCiCeg/.x/i

0i;jm

D.2x² 1/.^mC21/

m

Y

lD1

.S_{kl 1}.x//^{mC1 l} Y

1ijm

S_{k.ei dj/ 1}.x/;

det

h 1 S_{sCk.nCiCj /}.x/

i

0i;jm

D. 2x²C1/.^mC21/Qm

iD0.S_{k.iC1/ 1}.x//^{2.m i /} Q

0i;jmS_{sCk.nCiCj /}.x/ ; det.Œ.SnCiCj.x//^hmi_{0i;jd 1}/

D. 1/^d.^d₂/C.^dC₃¹/. 2x²C1/.^d2/

d 1

Y

iD1

Si 1.x/Sm i.x/d i

2.d 1/

Y

iDd 1

.SnC1.x//^{hmC1 di}: By Favard’s theorem [6] (see also the standard reference textbook by Chihara [5, Chapter 2]), the sequence P.x/.1; q; rI1; b; c/D.Pn/n2Z0 forms a sequence of orthogonal polynomials (with respect to certain linear functional) forq¤0andc¤0.

By Theorem2, Theorem3, Theorem4and Theorem5, we state some determinant identities for matrices containing (powers of) such orthogonal polynomials.

Corollary 6. Let n0, m 1 and d 1. Let .Pn/_n2Z0 be a sequence of orthogonal polynomials of the form:

P01; P1qxCr; PnC2D.xCb/PnC1CcPn

wherec¤0,q¤0andr; bare any real numbers. Then det.ŒP_n^m_CiCj0i;jm/

D. 1/^.nC1/.^mC₂¹/.^mC21/cⁿ.^mC₂¹/C2.^mC₃¹/

m

Y

iD0

m i

!

U_i^{2.m i /}_C1

;