GENERATING FUNCTIONS OF THE PRODUCT OF 2-ORTHOGONAL CHEBYSHEV POLYNOMIALS WITH SOME NUMBERS AND THE OTHER CHEBYSHEV POLYNOMIALS Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2021.8530

UDC 517.58, 517.52, 512.62

H. MERZÜÜK, B. ÄLÜUI, A. BOUSSAYOUD

GENERATING FUNCTIONS OF THE PRODUCT OF 2-ORTHOGONAL CHEBYSHEV POLYNOMIALS WITH SOME NUMBERS AND THE OTHER CHEBYSHEV POLYNOMIALS

Abstract. In this paper, we give the generating functions of binary product between 2-orthogonal Chebyshev polynomials and k-Fibonacci, k-Pell, k-Jacobsthal numbers and the other orthogonal Chebyshev polynomials.

Key words: 2-Orthogonal Chebyshev polynomials; k-Fibonacci; k-Pell and k-Jacobsthal numbers; Generating functions; Chebyshev polynomials

2010 Mathematical Subject Classification: 33C45, 42C05

1. Introduction. The Chebyshev polynomials Tn(x), Un(x), Vn(x) and Wn(x), n ^ 0, of the first, second, third and fourth kinds are respectively defined by the following formulas:

Tjcos 0) = cos(n0), Ujcos 0) = sin[(n + 1)0],

sin 0

T_, m cos(n +1/2)0 sin(n +1/2)0

Vn(COs 0) = -V ( '' , W„(cos 0)= \ ' ,

cos(0/2) sin(0/2)

where x = cos0, 0 E [0,n]. (For more details see [9], [10]).

The resulting polynomials, Tn(x), Un(x), Vn(x) and Wn(x) are multi-

' n V (-1 -1)

ples of the Jacobi polynomials. In fact, Tn(x) = Pn 2' 2 (x), Un(x) = (1 1) (-1 1) (1 -1) = Pn2'2)(x), Vn(x) = Pn 2'2)(x), Wn(x) = Pn2' 2)(x), n ^ 0, where

{pn«>^)}n^Q, (a, p = —m, a + P = — m — 1, m ^ 1), is the Jacobi polynomials given by the following explicit expression [7], [15]

P(a, 0) (x) = ^ fn) 2n-Vr(n + a + P + V + 1)r(n + P +1) ( 1)V Pn (x) ¿0W r(2n + a + P + 1)r(v + P + 1) (x 1) •

© Petrozavodsk State University, 2021

There is the following simple relations between the Chebyshev polynomials

Tn+1(x) = (n + 1)U„(x),n ^ 0. K(x) = U„(x) - Un-i(x), n ^ 0. W„(x) = U„(x) + Un-i(x), n ^ 0.

It is well known that Chebyshev polynomials are orthogonal, symmetric and satisfy the following Three-Term Recurrence Relation (TTRR)

Pn+2(x) = xP„+l(x) - Yn+lPn(x), n ^ 0, (1)

with initial conditions P0(x) = 1, P1(x) = x.

In the following table, we give for n ^ 0, the explicit expression of the parameters involved in (1) of the monic Chebyshev polynomials (for more details, see [11], [12]).

Pn(x) TTRR Initial conditions

Tn (x) Tn+2(x) = xTn+1(x) 4 Tn(x) T0(x) = 1, Ti(x) = x

Un(x) Un+2 (x) = xU„+i(x) - 1 Un(x) U0(x) = 1, Ui(x) = x

Vn(x) Vn+2(x) = xVn+i(x) 24 Vn(x) V0(x) = 1, Vi(x) = x

Wn(x) Wn+2(x) = xWn+i(x) - 1 Wn(x) W0(x) = 1, Wi(x) = x

Table 1: Some characteristic of Chebyshev polynomials.

It can be easily seen from the (1) that the generating functions for Tn(x), Un(x), Vn(x) and Wn(x) are respectively given by (see [9], [10], [13])

xt - £ Tn(x)tn, + t2 = £ Un(x)tn,

1 - 2xt + t2 ^ y ' ' 1 - 2xt + t2

n=0 n=0

1 - t ~ 1 + t ~

1 ' E V.(x)f, + t2 = E W.(x)t".

1 - 2xt + t2 ^ 1 - 2xt + t2

n=0 n=0

Now, we recall the notion of d-orthogonal polynomials. A remarkable characterization of the d-monic orthogonal polynomial sequence is that

those sequences satisfy a (d + 1)-order recurrence relation, which we write in the form

d-1

Pm+d+l(x) = (X - ^m+d) Pm+d(x) - ^ Y^-d-VPm+d-l-v(x), m ^ 0,

v=0

with the initial conditions P0(x) = 1, P-1(x) = 0 and if d ^ 2

n-2

Pn(x) = (x - £ra_i) Pn-i(x) - ^ Yn-l-VVPn-2-v(x), 2 ^ n ^ d,

v=0

and the regularity conditions Y^+i = 0, m ^ 0.

The 2-orthogonal monic Chebyshev polynomial (2-classical) of first the kind {Tn}n^0 studied in [8], and defined by the next relations where a and Y are constants (see also [14])

T0(x) = 1, T1 (x) = x, T2(x) = x2 — a Pn+3(x) = xTrn+2(x) — aTn+i(x) — YTn(x), n ^ 0, y = 0.

Definition 1. An d-orthogonal polynomial sequence {Pn}n^0 is called d-classical d-orthogonal polynomial sequence if both {Pn}n^0 and its derivative {Pn}n^0 are d-orthogonal.

Note that is the 2-classical 2-orthogonal polynomial sequence analou-gous to the Chebyshev orthogonal polynomial sequence of the first kind {Tn}n>0 (see [8]).

Lemma 1. [1]. For n G N, the generating function of the monic 2-orthogonal Chebyshev polynomial sequence is given by

1

(x) zn

n=0

1 - xz + az2 + yz3

In this paper, we use the new generating function of the 2-orthogonal monic Chebyshev polynomial sequence (2-classical) to give some new generating functions related to the product of the 2-orthogonal Chebyshev polynomials and k-Fibonacci numbers, k-Pell and k-Jacobsthal numbers. We also, use this to derive some new symmetric properties of the generating function of the product of the 2-orthogonal Chebyshev polynomials and other Chebyshev polynomials.

2. Generating and symmetric functions. Now, we need to introduce a new symmetric function and we give some properties related to this function. We also, give some more useful definitions from the literature which are used in the subsequent sections.

We shall handle functions on different sets of indeterminates (called alphabets, though we shall mostly use commutative indeterminates for the moment). A symmetric function of an alphabet A is a function of the letters which is invariant under permutation of the letters of A. Taking an extra indeterminate z, one has two fundamental series

Az (.4) = n (1 + <A) = nobs) •

«eA 11v '

«eA

The expansion of which gives the elementary symmetric functions An(A) and the complete functions Sn(A) :

Az(A) = Y An(4)zn, ^z(A) = Y Sn(A)zn

n=0 n=0

Start with the following definitions.

Definition 2. Let A and B be any two alphabets, then we give Sn(A — B) by the following form

n (1 — zb) x

nB(i _ zs) = Y Sn(A — B)zn = az(A — B), (3)

A ( ) n=0

«eA

with the condition Sn(A — B) = 0 for n < 0 [2]. Taking A = {0, 0, • • •, 0} in (3) gives

J](1 — *&) = £ Sra(—B)zn = Az(—B).

beB n=o

Further, in the case A = {0, 0,..., 0} or B = {0, 0,..., 0}, we have

X

Y Sn(A — B)zn = az (A) x Az (—B ).

n=0

Thus,

n

S„(A - B) = Y Sn-fc(A)Sfc(-B). (6)

fc=0

Definition 3. [3] Let g be any function on Rn, then we consider the

divided difference operator as the following form

(g) =

g(x1 , . . . , , xi+1 , . . . , xn) g(x1 7 . . . 7 xi-1 7 Xi+1 7 7 Xi+2 7 ... 7 Xn)

Definition 4. [4] Given an alphabet E = {e1 )e2} ; the symmetrizing operator #kie2 is defined by

efc+n _ efc+n

<%e2 (en) = --— = Sfc+n-1(e1 + —2) M ^ 0.

12 -1 - -2

In this part, the following lemmas is one of the key tools of the proof of our main results.

Lemma 2. [2] Given an alphabet E = {-1;-2; —3} ; we have

YSn (E) zn = --—-—-T 7 (7)

n=0 (1 - — 1z) (1 - —2z) (1 - —3z)

with

(1 - —1Z) (1 - — 2Z) (1 - —3Z) =

= 1 - (—1 + —2 + — 3)z + (—1—2 + —1—3 + — 2—3)z2 - —1—2—3Z3 =

= 1 + S1 (-E) z + S2 (-E) z2 + S3 (-E) z3.

Lemma 3. [2] Given two alphabets E = {—1; —2) —3} and A = {a1; -a2}, we have

^ c i a\ n - 1 + a1 ^2 (-E) z2 + a^a - 0^3 (-E) z3

n=0 Sn(E)Sn(A)z = n (1 - —M n (1 + —02z) .

n=0 e€E e€E

Lemma 4. [2] Let A = {a1; -a2} and E = {—1;—2)—3} two alphabets, we have

Y Sn (E )Sn-1(A)zn =

n=0

= -Si (-E) z - (ai - a2)S2 (-E) z2 - ((ai - a2)2 + ai^2)^3 (-E) z3

n (1 - eaiz) n (1 + ea2z)

r Si(-E) = -x

Note that, the substitution of < S2(-E) = a , in (7) gives the fol-

{ S3(-E) = 7

lowing result:

ro +ro

£ S„(E)zn = 1-+ 2 + 3 = £ Tpn(x)zn,

1 — xz + az2 + yz 3 '

n=0 n=0

which represents the generating function of 2-orthogonal monic Chebyshev polynomials, with Sn(E) = Tn(x), (see [1]).

3. Generating functions of binary products of 2-Chebyshev polynomials and numbers. In this section, we are going to create the new generating functions of products of 2-Chebyshev polynomials and some numbers (k-Fibonacci, k-Pell and k-Jacobsthal) based on Lemmas 2, 3 and 4.

Theorem 1. For n G N, the generating function of the product of 2-orthogonal Chebyshev polynomial and k-Fibonacci numbers is given by

£ TB(x)iUzn =1 + af(+ kYz3 ,

n=0 fi(z)

with

fi(z) = 1 - kxz + ((k2 + 2)a - x2)z2 + ((k2 + 3)kY + kax)z3 + + ((k2 + 2)yx + a2)z4 + kaYz5 - y2z6.

Proof. By [6], we have Fk;n = Sn (ai + [-a2]). Then, we can see that

ro ro

£ TT„(x)Ffc;„zn = £ Sn (E) Sn (ai + [-a2]) zn =

n=0 n=0

ro ro

— ^i £ Sn(E)(aiz)n + a2 £ Sn(E)(-a2z)n)

(ai + a2^ n=0 n=0

In addition, we have

ro1 n=0 Sn (E) (aiz)n = 1 + Si(-E)aiz + S2(-E)a2z2 + S3(-E)a3z3,

and

(E) (-Mn = 1 - 5l(-E)a2z + S2(-E)a2z2 - &(-£)a2z3' According to Lemma 2, we obtain

"nQrn(x)Fk,nzn ai + + + ^(-E )a2z2 + Ss(-E )aiz3 +

«2

+

1 - Si(-E)«2Z + S2(-E)«2z2 - Sa(-E)«3z3 Then, by reduce to same denominator, we obtain the following result

T ()F n =_1+ Pi(x)z2 + P2(x)z3

2^Tn(x)Fk,nz (T)z + ^(T)

n=0

where

1+qi(x)z + q2(x)z 2 + q3(x)z 3 + q4(x)z4 + q5(x)z 5 + q6(x)z6;

pi(x) = a^S^-E), p2(x) = aia2(ai-a2)S3(-E), qi(x) = (ai-a2)Si(-E),

q2(x) = S2(-E)(ai - a2)2 - a^Si-E)2 - 2S2-E)),

?3(x) = S3(-E)(ai - a2)3 - a^fai - a2)(Si(-E^(-E) - 3^-E)),

&(x) = -aia2(ai-«2)^3 (-E )Si(-E )+ai«2(S2(-E )2 - 2S3(-E)Si(-E)),

q5(x) = a^S^-E)S2(-E)(ai - a2), q6(x) = -S3(-E)2aia2-

After a simple calculation, of p^(x) and ^(x), we obtain the desired result. □

Theorem 2. For n G N, the generating function of the product between 2-orthogonal Chebyshev polynomials and k-Pell numbers is given by

V T (r)p zn = xz - 2az2 - y (4 + k) z3

/ Tn(x)P k,nz f f \ ,

n=0 f2(z)

with

f2(z) = 1 - 2xz + (2(k + 2)a - kx2)z2 + (2y(3k + 4) + 2kax)z3 + + (2k(k + 2)yx + k2a2)z4 + 2k2aYz5 - k3Y2z6.

Proof. By referred to [6], we have Pk,n = Sra-i (ai + [—«2])- On the other hand, we can see that

ro ro

£ T„(x)Pfc,„zn = £ Sra (E) Sn_i (ai + [—«2]) zn =

n=0 n=0

ro ro

-L t X ^ / „s / sra

(ai + ß2)

( £ S„(E)(aiz)n - £ Sn(E)(-a2z)n

n=0 n=0

Using the Lemma 3, we can write

n 1

n=0Tn(x)Pk,nZ ai + a2 V1 + Si(-E)aiz + S2-E)a2z2 + S3-E)a?z3

1 \

(8)

1 - Si(-E)a2z + S2(-E)a2z2 - Ss(-E)a2z3

Equivalently

£fn(x)P

k

n_ Pi(x)z + P2 (x)z2 + P3(x)z3

1 + 9i(x)z+q2(x)z2+q3(x)z3 + q4(x)z4 + q5(x)z5 + q6(x)z

n=0

where

pi(x) = — Si(-E), P2(x) = —(ai — a2)S2(—E), P3(x) = —((ai — a2)2 + aia2)S3(—E), qi(x) = (ai — a2)Si (—E), 92 (x) = S2(—E )(ai — a2)2 — aia2(Si(—E )2 — 2S2(—E)), 93 (x) = S3 (—E) (ai — a2)3 — aia2(ai — a2)(Si (—E) S2 (—E) — 3S3 (—E)), q4(x) = aia2(ai — a2)2S3(—E)Si(—E) — aia2(S2(—E )2 — 2S3(—E )Si(—E)),

95(x) = aia2S3(—E)S2(—E)(ai — a2), 96(x) = —S3 (—E)2 aia^ This gives, after a simple calculation, the following

pi(x) = x,p2(x) = — 2a,p3(x) = —7(4+k), qi(x) = — 2x, q2(x) = 2(k+2)a—kx2,

93(x) = 27(3k + 4) + 2kax, q4(x) = 2k (k + 2) yx + k2a2, 95 (x) = 2k2aY, 96 (x) = —k3 y2. Hence, the Theorem 2 is valid- □

Theorem 3. For n G N, the generating function of the product of 2-orthogonal Chebyshev polynomials and k-Jacobsthal numbers is given by

— - xz — kaz2 — y (2 + k2) z3

^ ^ Tn (x) Jk, nz

n=0

with

f3(z)

f3(z) = 1 - kxz +((k2 + 4) a — 2x2) z2 + (kY(k2 + 6) + 2kax) z3 + + (2 (k2 + 4) yx + 4a2) z4 + 4kaYz5 - 8y2z6.

Proof. Recall that, we have Jk,n = Sn-i (ai + [—a2]) (see [6]). We see that

x

YTn(x)Jk,nzn = Y Sn (E) Sn-1 (ai + [—a2]) z

n=0 n=0

x x

n

( Y Sn(E)(aiz)n — Y Sn(E)(—a2z)n

(ai + a2)V^—

n=0 n=0

According to Lemma 4, this gives the following equality

y^Tn(x)J/

x)Jk,nz _

n=0 ai + a2 V1 + Si(—E)aiz + S2—E)a2z2 + S3—E)a3z3

1

1 — Si(—E)a2z + S2(—E )a2z2 — S3(—E )a2z3/'

Then, by reduce to same denominator, we get

VT (x)J zn =_pi(x)z + P2(x)z2 + P3(x)z3_

n^ n ,n 1+ 9i(x)z + q2(x)z2 + q3(x)z3 + q4(x)z4 + q5(x)z5 + q6(x)z6;

where

pi(x) = —Si(—E), P2(x) = —(ai — a2)S2(—E), P3(x) = —((ai — a2)2 + aia2)S3(—E), qi(x) = (ai — a2)Si(—E), q2(x) = S2(—E )(ai — a2)2 — aia2(Si(—E )2 — 2S2(—E)), 93 (x) = S3(—E )(ai — a2)3 — aia2(ai — a2)(Si(—E )S2(—E) — 3S3(—E)), q4(x) = aia2(ai — a2)2S3(—E )Si(—E) — a?a2(S2(—E )2 — 2S3(—E )Si(—E)),

1

1

95(x) = ala^-E)(ai - a2), 96 (x) = — S3(—E)2aia2. This gives, after a simple calculation, the following

p1(x) = x, p2(x) = —ka, p3(x) = —y(2 + k2), q1(x) = —kx,

92 (x) = (k2 + 4)a — 2x2, q3(x) = kY(k2 + 6) + 2kax,

q4(x) = 2(k2 + 4)yx + 4a2, q5(x) = 4kaY, q6(x) = —8y2.

Hence, the Theorem 3 is valid. □

4. Generating functions of binary products of 2-Chebyshev polynomials and other Chebyshev polynomials. The following lemmas (see [2]) are the key tools of the proof of our main results.

Lemma 5. Given two alphabets E = {e1, e2, e3} and A = {2a1, — 2a2}, we have

^ c fi?\Q t a\ n - 1 + 4aia2S2 (—E) z2 + 8aia2(ai — a2)^3 (—E) z3 Sn(E)Sn(A)Z = n (1 — 2eaiz) n (1 + 2ea2Z) .

"=0 e€E e€E

Lemma 6. Let A = {2ai, — 2a2} and E = {ei,e2,e3} two alphabets, we have

£ Sra(E)Sra-i(A)zra =

n=0

= —Si (—E) z — 2(ai — a2)S2 (—E) z2 — 4((ai — a2)2 + a^)S3 (—E) z3

= ¡I (1 — 2eaiz) n (1 + 2ea2z) .

Based on the last lemmas, we can state the following theorems which represent the new generating functions of products of 2-Chebyshev polynomials of the first kind with the other Chebyshev polynomials.

Theorem 4. For n G N, the generating function of the product of 2-orthogonal Chebyshev polynomials and monic chebyshev polynomials of the second kind is given by

1 — az2 — 2Yyz3

£fra(x)Ura(y)zn =-—)

n=0 f4(z)

with

/4(z) = 1 - 2xyz + (2 (2y2 - 1) a + x2) z2 + (2(4y2 - 3)Yy - 2axy)z3-

- (2(2y2 - 1)yx - a2)z4 + 2aYyz5 + y2z6.

Proof. By using [5], we have Un(y) = Sn (2ai + [-2a2]). Then, we can easily seen that

x

jn

Y TT„(x)U„(y)zn = £ Sn (E) Sn (2ai + [-2a2]) z'

Jn ) ^n

n=0 n=0

x

2(ai + a,Sn(E)(2aiz)n + 2a2 ^ Sn(E)(-2a2z)

n=0 n=0

1 2a!

2 (ai + a2j 1 + 2Si(-E)aiz + 4S2(-E)a2z2 + 8S3(-E)a3z3

+_2a_)

+ 1 - 2Si(-E)a2z + 4S2(-E)a2z2 - 8S3(-E)a3z3^ by using the Lemma 5. Equivalently, we get

V T^n(x)Un(y)zn =_1 + fi(x)z2 + f2(x)z 3_,

n^ n n 1+gi(x)z + g2(x)z2 + g3(x)z3 -g4(x)z4 + g5(x)z5 -g6(x)z6'

where

/i(x) = 4aia2S2(-E), /2(x) = 8aia2(ai-a2)S3(-E), gi(x) = 2(ai-a2)Si(-E),

g2(x) = 4S2 (-E) (ai - a2)2 - 4aia2(Si (-E)2 - 2S2 (-E)),

g3(x) = 8S3 (-E) (ai -a2)3-8a^(ai -a2)(Si (-E) S2 (-E) -3S3 (-E)),

g4(x) = 16aia2(ai-a2)2S3-E )Si(-E )-16a2a2(S2(-E )2-2S3(-E)Si(-E)),

g5(x) = 32aia2S3 (-E) S2 (-E) (ai - a2), gj(x) = 64S3 (-E)2 a3a2.

After a simple calculation of /¿(x) and gj(x) we obtain the desired result. □

Theorem 5. For n G N, the generating function of the product of 2-orthogonal Chebyshev polynomials and Chebyshev polynomials of the first kind is given by

+x 1 I „.(Or, 2 1 \ -v2 , (Ar.,2 Q\ „,^3

,Tn(x)Tn(y)z'

n=0

n

1 - xyz + a (2y2 - 1) z2 + y (4y2 - 3) yz3

nn

^ f5(z)

(I

with

f5(z) = 1 — 2xyz + (2(2y2 — 1)a + x2)z2 + (2(4y2 — 3)Yy — 2axy)z3 — — (2(2y2 — 1)yx — a2)z4 + 2aYyz5 + y2z6.

Proof. By [5], we have Tn(y) = Sra (2ai + [—2a2]) — ySra_i (2ai + [—2a2]). In addition, we can see that

ro

£ Tra(x)Tra(y)zra =

n=0

ro

£s„(E )(Sn(2ai + [—2a2 ]) — ySra_i(2ai + [— 2a2]))zn =

n=0

roro

= £ Sn(E)Sn(2ai + [—2a2])zn — y £ Sra(E)Sra_i(2ai + [—2a2]) =

n=0 n=0

ro ro ro

= £ Tra(x)Ura(y)zra— 2( + ) (£ Sn(E)(2aiz)n—£ Sra(E)(—2a2z) j ,

n=0 ^ ' n=0 n=0

which gives

roro

£ TTn(x)Tra(y)zra = £ Tra(x)Ura(y)zra—

n=0 n=0

__y_f_1__

2 (ai + a2^1 + 2Si(—E)aiz + 4S2(—E)a?z2 + 8S3(—E)a3z3

1 — 2Si(—E )a2z + 4S2(—E )a2 z2 — 8S3(—E )a3z3, After reduce to same denominator, we obtain the following result

ro

jn

£Tra(x)Tra(y)zra = £ Tra(x)Ura(y)zr

n=0 n=0

fi(x)z — f2(x)z2 — f3(x)z3

.1 + gi(x)z + g2(x)z2 + g3(x)z3 — g4 (x)z4 + g5(x)z5 — g6(x)z6 where

fi(x) = — Si(—E), f2(x) = 2(ai — a2)S2(—E), fs(x) = 4((ai — a2)2 + a^^—E ),gi(x) = 2(ai — a2)Si(—E),

1

g2(x) = 4S2(-E)(ai - a2)2 - 4aia2(Si(-E)2 - 2S2(-E)),

g3(x) = 8S3(-E)(ai - a2)3-8a^(ai - a2)(Si(-E)S2(-E)-3S3(-E)),

g4(x) = 16aia2 (ai-a2)2S3-E )Si(-E )-16a2a2(S2(-E )2-2S3(-E)Si(-E)),

g5(x) = 32a?a2S3(-E)S2(-E)(ai - a2),^(x) = 64S3(E)2a3a2.

After a simple calculation of /¿(x) and gj(x) we obtain the result. □

Theorem 6. For n G N, the generating function of the product between 2-orthogonal Chebyshev and Chebyshev polynomials of the third kind is given by

+x 1 - xz + a (2y - 1) z2 + y (4y2 - 2y - 1) z3

^Tn(x)Vn(y)zn = 1 - + a (2y - / ( +

n=0 /6(z)

n=0

with

/6(z) = 1 - 2xyz + (2 (2y2 - 1) a + x2) z2 + (2 (4y2 - 3) Yy - 2axy) z3-

- (2 (2y2 - 1) yx - a2) z4 + 2aYyz5 + y2z6.

Proof. By referred to [5], we have

Vn(y) = Sn (2ai + [-2a2]) - Sn-i (2ai + [-2a2]).

By using the Lemma 6, we obtain

x x

n

]>]Tn(x)Vn(y)zn= J]Sn(E)(Sn(2ai + [-2a2]) - Sn-i(2ai + [-2a2]))zn =

n=0 n=0

xx

= Y Sn(E)Sn(2ai + [-2a2])zn - Y Sn(E)Sn-i(2ai + [-2a2]) =

n=0 n=0

x x x

= Y T^n(x)Un(y)zn- 2( 7 Y Sn(E)(2aiz)n-Y Sn(E)(-2a2z)n).

n=0 i 2 n=0 n=0

Equivalently

xx

Y T^n(x)Vn(y)zn = Y Tn(x)Un(y)zn-

n=0 n=0

__(_1__

2(a, + a2^ 1 + 2Si(-E )a, z + 4S2 (-E )aiz2 + 8S'3(-E )a3z3

1 - 2Si(-E)a2z + 4S2(-E)a2z2 - 8Sa(-E)a3z3, which gives, after reduce to same denominator, the following result

£T„(x)Vn (y)zn = £ TTra(x)Ura(y)zn

n=0 n=0

fl(x)z - f2(x)z2 - f3(x)z3

1 + gl(x)z + g2(x)z2 + g3(x)z3 - g4(x)z4 + g5(x)z5 - g6(x)z6 '

where

/i(x) = -Si(-E), f2(x) = 2(ai - a2)S2(-E), /3(x) = 4((ai - a2)2 + aia2)S3(-E), gi(x) = 2(ai - a2)Si(-E), g2(x) = 4S2 (-E) (ai - a2)2 - 4aia2(Si (-E)2 - 2S2 (-E)), g3 (x) = 8S3(-E )(ai-a2)3 - 8a^(ai - a2)(Si(-E )S (-E) - 3S,(-£)), g4(x) = 16aia2(ai - a2)2Si(-E )Si(-E) - 16a2a2(S2-E )2 - 2S3-E )Si(-E)), g5(x) = 32a2a2S3(-E)&(-£)(ai - a2),gj(x) = 64S3(-E)2a3a2.

Hence, after a simple calculation of /¿(x) and gj(x), we can obtain the result. □

Theorem 7. For n G N, the generating function of the product between 2-orthogonal Chebyshev and Chebyshev polynomials of the fourth kind is given by

£ f. (x)w.(y>z- = 1+xz - a (2y + Y (4y2 + 2y -11 z3,

t^k f7(z)

with

/7(z) = 1 - 2xyz + (2(2y2 - 1)a + x2)z2 + 2(4y2 - 3)Yy - 2axy)z3-

- (2(2y2 - 1)yx - a2)z4 + 2aYyz5 + y2z6.

Proof. According to [5], we have

Wn(y) = Sn(2ai + [-2a2]) + Sra_i(2ai + [-2a2]).

1

We, easily, see that

x x

^Tra(x)Wra (y)zn = )(Sra(2ai + [-2a2]) + S„-1(2a1 + [-2a2]))zn

Jn

n=0 n=0

oo oo

= Y Sn (E) Sn (2ai + [-2a2]) zn + Y Sn (E) Sn-i (2ai + [-2a2]) =

n=0 n=0

x x x

= Y T^n(x)Un(y)zn + + A Y Sn(E)(2aiz)n- Y Sn(E)(-2a2*) j,

n=0 n=0 n=0

which implies that

xx

Y Tn(x)Wn(y)zn = Y T^n(x)Un(y)zn+

n=0 n=0

2(ai + a2^1 + 2Si(-E )aiz + 4S2(-E )a2z2 + 8Ss(-E )a?z3

1

1 - 2Si(-E)a2Z + 4S2(-E)a2z2 - 8S3(-E)a3z3

since we have the Lemma 6. Then, by reduce to same denominator, we get

xx

Y ?n(x)Wn(y)zn = Y ?n(x)Un(y)zn+

n=0 n=0

+_/i(x)z - /2 (x)z2 - f3(x)z3_

1 + gi(x)z + g2(x)z2 + g3(x)z3 - g4(x)z4 + g5(x)z5 - ge(x)z6 '

where

/i(x) = -Si(-E), /2(x) = 2(ai - a2)S2(-E),

/3(x) = 4((ai - a2)2 + aia2)S3(-E), gi(x) = 2(ai - a2)Si(-E),

g2(x) = 4S2 (-E) (ai - a2)2 - 4aia2(Si (-E)2 - 2S2 (-E)),

g3(x) = 8S3(-E)(ai - a2)3 - 8aia2(ai - a2)(Si (-E) S2 (-E) - 3S3 (-E)),

g4(x) = 16aia2 (ai-a2)2S3 (-E )Si(-E )-16a2a2(S2(-E )2 - 2S3-E)Si(-E)),

g5(x) = 32a2a2S3(-E)S2(-E)(ai - a2), g6(x) = 64S3(-E)2a3a2.

This gives, after a simple calculation of /¿(x) and gj(x), the desired result. □

1

Conclusion. In this paper, the new theorems has been proposed in order to determine the generating functions. The proposed theorems is based on the symmetric functions. The obtained results agree with the results obtained in some previous works.

Acknowledgment. Sincere thanks are due to the referee for his/her careful reading of the manuscript and for his/her valuable comments. This work was supported by Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.

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Received May 13, 2020. In revised form, November 10, 2020. Accepted November 16, 2020. Published online December 16, 2020.

H. Merzouk

LMAM Laboratory and Department of Mathematics Mohamed Seddik Ben Yahia University, Jijel, Algeria merzoukhind07@gmail.com

B. Aloui

Gabes University, Higher Institute of Industrial Systems of Gabes

Department of Electromechanics

Street Salah Eddine Elayoubi 6033 Gabes, Tunisia

Baghdadi.Aloui@fsg.rnu.tn

A. Boussayoud

Laboratory and Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria alboussayoud@gmail.com