Q-CHEBYSHEV POLYNOMIALS AND THEIR Q-CLASSICAL CHARACTERS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 11 (29), No 1, 2022, pp. 81-101 81

DOI: 10.15393/j3.art.2022.10330

UDC 517.58, 517.518

M. Mejri

g-CHEBYSHEV POLYNOMIALS AND THEIR ^-CLASSICAL CHARACTERS

Abstract. In this work, we give some properties of the g-Chebyshev polynomials through the Stieltjes function associated with their regular forms (linear functional). Some connection formulas are highlighted. The integral representation of those forms are given.

Key words: q-difference equation, Hq-semiclassical polynomials, orthogonality measure

2020 Mathematical Subject Classification: Primary: 33C45; Secondary: 42C05

1. Introduction. In this contribution, we introduce the monic orthogonal polynomial sequence (MOPS) of g-Chebyshev polynomials [Tn(x, q)}n^o of the first kind and {Un(x, q)}n^o of the second kind, which are orthogonal with respect to the forms (linear functionals) Tq, Uq, respectively, through a ^-difference functional equation similar to that satisfied by the classical Chebyshev forms T and U of the first/second kind, respectively [15]. In addition, we preserve the connection property Tq11 = Uq, where Tq11 is the first associated form of Tq. Note that many authors have been interested in the ^-extension of the Chebyshev polynomials and their properties [1], [4], [8]. The normalized sequences associated with those introduced in [1], [4], [8] are equal to {Tn(x,q)}n^0 and {Un(x, q)}n^o, respectively, up to a dilation. Furthermore, those polynomials are a particular cases of big g-Jacobi polynomials [1], [9]. Our main aim is to study in detail these polynomials through their ^-classical character. The second section is devoted to the preliminaries, some fundamental results useful in the sequel, to the introduction of the MOPSs {Tn(x, q)}n^o and {Un(x, q)}n^o. In the third section, we obtain a connection formula between Tq and the shifted form Uq, which is a ^-extension of the formula given in [10]. As a consequence, we highlight certain formulas connecting

© Petrozavodsk State University, 2022

the polynomials Tn(x,q) and Un(x,q) for n ^ 0, which are g-extensions of the classical case [3], [16], [17]. In the fourth section, we give the integral representation of the form Tq using the connection formula given above. This has not been done previously in literature. In the last section, we give explicitly the Stieltjes functions of the g-Chebyshev form of the first and second kind.

2. Preliminaries and fundamental results. Let V be the vector space of polynomials with coefficients in C, and let V be its dual. We denote by (w, f) the effect of w E V on f E V. In particular, we denote by (w)n: = (w,xn), n ^ 0 the moments of w. Let us introduce some useful operations in V'. For any linear form w, any polynomial g and any a E C — {0}, c E C, let gw, w', haw, (x — c)-1w, 8C, and Hqw be the forms (linear functionals) defined by duality

(9w,f) = (w,gf), (w'J) = —(w,f), (hawJ) := (w,haf),

((x — c)-1w, f): = (w, ecf), (Sc, f): = f (c), (Hgw, f) = —(w, Hqf ),f E V,

where

(Bcf )(X) = f (X) — f (C), (haf )(X) = f (ax), x — c

Hq(f )(X) = f W-fM ,x = 0,q E C — ({0} U □ {z E C,zn = 1}) , ^ )X

Hq (f )(0) = f (0).

We also define the right-multiplication of a form by a polynomial as

(wf )(x): = (w,-—-j,w EV ,f EV.

\ x— q /

The Stieltjes function of w eV is defined by

S(z ,w) =

n>0

Zn+l

A monic polynomial sequence (MPS) {Pn}n^o is a sequence of monic polynomials Pn, n ^ 0, with degPn = n. Let {wn}n^0 be its dual sequence, defined by (wn, Pm) = 8n,m,n,m ^ 0. The MPS {Pra}ra^o is orthogonal (MOPS) with respect to w E V if the following conditions hold: (w,PmPn) = rn8n>m, n,m ^ 0, rn = 0, n ^ 0. In this case, the form w

is said to be regular. The form w is called normalized if (w)0 = 1. In this paper, we suppose that all forms are normalized. Thus, w = w0 and [Pn}n^o satisfies the standard recurrence relation

Po(X) = 1 , Pi(x) = x - Po,

Pn+2(x) = (X - Pn+l)Pn+l(x) - ln+iPn(x), 7n+i = 0,n ^ 0.

The regular form w0 is said to be symmetric if (w0)2n+l = 0, n ^ 0 or, equivalently, = 0, n ^ 0 in (1) [3], [11].

Let {Pn}n^0 be the sequence defined by Pn(x) = a-nPn(ax), n ^ 0, a = 0. It is a MOPS with respect to w0 = ha-iw0 fulfilling (1) with [10]

a ln+1 ^ n Pn = -, ln+1 = -2T- ,n ^ 0.

a a2

Given a regular form w and the corresponding MOPS {Pn}n^0 satisfying (1), we define the first associated sequence {P^1}^ by [11]

Pni](x) = UPn+l(x) - P-+l(i)) = (w9oPn+l)(x). \ x- % /

{Pn^n^o is a MOPS with respect to w{1) satisfying (1) with [3], [11]

Pn} = pn+l, l{n+1 = 1n+2, n ^ 0.

The Chebyshev MOPS of the first kind (respectively, of the second kind), orthogonal with respect to T (respectively, U) are defined by [3], [11], [12]

{

!

T0(x) = 1, Tl(x) = x,

Tn+2(x) = xTn+l(x) - -yT+Mx), n ^ 0,

((x2 - 1)T)' - xT =0,

11 7i = 2, M = 4 ,n > 1

+1

(T,f) = - i ^=L= f (x)dx, f eV. K J V1 - x2 -l

U0(x) = 1, Ul(x) = x,

Un+2(x) = xUn+l(x) - m+-lUn(x), n ^ 0,

u 1

ln+i = 4 ,n > 0,

((x2 - 1)U)' - 3xU = 0, +i

(u, f) = 2 j ^Ï—Xx2f (x)dx f eV. (2)

-i

Let us recall some results:

Lemma 1. [7], [11] Let w eV ', f eV, a e C - {0}. The following formulas hold:

Hq (fw) = (h— f)Hq (w) + q-1(Hq-i f)w, (3)

5 (z Jw) = f(z)S (z ,w) + (wdof )(z), (4)

ha( fw) = (ha-i f)(haw). (5)

Definition 1. [7] A form w is called Hq-semiclassical, if it is regular

and if there exist two polynomials ( and ^ (( is monic), deg( = t ^ 0, deg-0 = p ^ 1, such that

Hq ((w)+^w = 0, (6)

the corresponding orthogonal sequence {Pn}n^o is called Hq-semiclassical.

Remark. When deg( ^ 2, deg^ = 1, w is called Hq-classical form [6].

Lemma 2. [7] If w is Hq-semiclassical, fulfilling the equation (6), the form w = ha-iw, a e C — {0} is Hq-semiclassical and satisfies

Hq ((w) + rtpw = 0,

with ((x) = a-degax), ip(x) = a1-deg^(ax).

The Hq-semiclassical character of a regular form can be described via the formal Stieltjes function, as follows.

Theorem 1. [7] Let w be a regular form. The following statements are equivalent:

(a) w is Hq-semiclassical form satisfying (6).

(b) The Stieltjes function S(^,w) satisfies the q-Riccati equation (hq-i4>)(z)Hq-i (S (z ,w)) = C (z)S (z ,w) + D(z),

where

C = -H-i® - qiP, D = -{(H— (wd o®) + q(w6 o4>)}. We are going to use the following notations and results [5]:

( 1, n = 0,

(a, q)n := I n—l

I II (1 - a<f), n > 1, a e C.

v u=0

lim (q^, = (1 - z)-a, |z| < 1, ae R. (7)

i^1 (z, q)<x>

Let {Pn( a)}n>0 be the symmetric MOPS introduced in the situation (3.25) in [15] with q is replaced by q2

7 +1(a) = (qn+1 - 1)(qn+l+2a - 1) qn+2«+2 n > 0

^n+l(a) = (q2n+2a+l - 1)(q2n+2a+3 - rf ,U > 0

1 - g-2a-2 (8)

H„((x2 - 1)u(a)) +--xu(a) = 0.

1-

If a = -1, we denote Tq: = u(-1), %++]_: = ^n+l(-1), n > 0; then

% = q % = qn+1 > 1

71 q+1 ,ln+1 (qn + 1)(qn+1 + 1),U >, (9)

Hq((X2 - 1)Tq) - q-lxTg = 0.

Note that if q ^ 1, we obtain the form T [3], [12]; then the form Tq is its ^-extension (we say that Tq is the g-Chebyshev form of the first kind). In the following, {Tn(x, q)}n>0 is the MOPS with respect to Tq.

Denote by Uq: = h- iu( 2,), ^fn+ l: = q-1 Jn+l( 2), n > 0 and obtain, due to (8) and Lemma 2:

n +2

1 = („n+l in {„n+2 + 1) ,n > 0,

1 - q-3 ~q

(qn+1 + 1)(qn+2 + 1V (10)

1 -3

Hq((x2 - q-1) Uq) + --—X Uq = 0.

If q ^ 1, we get the form U [3], [12]; then Uq is the ^-extension of the form U (it is the g-Chebyshev form of the second kind). Denote by {Un(x, q)}n^0 the MOPS with respect to Uq.

Remark. We see that, due to (9) and (10), we have

Til) = Uq.

This is a q-extension of the classical case [10].

3. Connection formulas.

Lemma 3. [14] Let (bn)n^0 with bn = 0, n ^ 0, (cn)n^0 be two sequences of complex numbers and (xn)n^0 be a sequence satisfying the following recurrence relation

Xn+i = bnXn + Cn,n ^ 0, xo = a e C - {0}.

We have Xn+i = ( n bk) {a + ^ ( n O °k\,n ^ 0.

V k=0 ' L k=0 \u=0 ' }

Lemma 4. The following equation holds:

(X2 - 1)Tq = - ^ h- 1 Uq. (11)

Proof. Let wq be the normalized form defined by

1

(X2 - 1)Tq = - —Wq. (12)

Then Hq ((X2-1)Tq) = - -+Hq (Wq), and from (9) we get XTq = - Hq (Wq). Multiplying both sides by x2 - 1 and using (12), we deduce that (x2 - 1) Hq(wq) - q-iXWq = 0.

From relation (3), it follows that

1 - q-3

Hq ((x2 - q 2)Wq) + —-—XWq = 0. (13)

On the other hand, from (10) and by virtue of Lemma 2, we have

1 — a-3

Hq{(x2 - q-2)h -2Uq) + ^—xh -2Uq = 0. (14)

q2 1 — q q 2

Based on relations (13), (14), and the fact that (h— iUq)0 = (wq)0 = 1,

we get h -1 Uq = wq, which provides (11). □

q 2

Remark. When q ^ 1 in relation (11), we obtain the connection formula given in [10].

In the sequel, we denote by {Un(x, q)}n>0 the MOPS with respect to U„ : = h _iUa. Then we have

!

Un(x, q) = q 2 Un(q2x, q),n ^ 0. (Jo(x, q) = 1, Ui(x, q) = x,

Un+2(x, q)(x, q) = xÙn+iix, q) - \Ùn(x, q), n ^ 0

:i5)

with

- qn+l

1 =1 + 1)1 + qn+2) > 0. (16)

Lemma 5. The following formulas hold:

2

(-1; q-l)n+l

n

Un(1, q) = ( -1) Y, ^, n > 0. (17)

k=0

Uilm-4 = 7^+1- ± 1-^> 0. (18)

Proof. From relations (15) and (16), it follows that

n +l

Un+2(1, q) = Un+l(1, q) - ^ + qn+l)(^1 + qn+2) (jn(1,1), n >

Equivalently,

(1 + qn+2)Un+2(1, q) - qn+2Un+l(1, q) = 1

((1 + qn+1)Ûn+i(1, q) - qn+1Un(1, q)) , n > 0.

1 + n +l

Therefore,

(1+,"+2)Un+2(1,-^Un+l!„) = {l + q)i1 q) - q(W'q), n > 0.

n (1 + <iM)

k=0

Since U0(1, q) = lJi(1, q) = 1, we get

1

1 + q

(n+l)(n+2)

1 T~T M \ 1 2

Un+i(1, q)= ^ -n-l^~n(1, q) + ^-, n > 1,

n(1 + Q-k ) k=0

but the previous relation is valid for n = 0. Using Lemma 3, we obtain

1 n

Un+i(1, 0) = n+2-+ * =

n (1 + q-k) k=0

k=1

n+1

2 ^ _fe(fe+1) ^ _ > q 2 , n ^ 0.

(-1; q-1)n+2

k=0

n

~ 2 fe(fe+i)

Thus, Un(1, q) = -,-77- / q 2 , n ^ 1. But the previous rela-

(-1; q-1)n+i

tion is valid for n = 0: this provides (17).

Based on relations (15) and (16), we get

(1 + qn+3)ùn%(1, Q) - Qn+3^~ni)i(1, q) =

-_1_(a r,n+2\U

((1 + qn+2)Unii(1, q) - qn+2Uni}(1, q)), n > 0.

1 + qni2 Then

(1 + qni3)U{n%(1, q) - qni3Û{nïi(1, Q) =

(1 + q2)(~~{1i)(1, q) - q2(~~(i)(1, q) 2(1 + q)

n n+2

n (1 + q k+2) n(1 + qk)

k=0 k=0

So,

-, n/i \ (n+2)(n+3)

eu,1) = T^rn!.)q- 2 , n > 1,

+ ' n(1 + r")

k=0

and, by Lemma 3, since the last relation is valid for n = 0, we get

T~T(iWi \ 2(1 + <i) V1 ~(fe+1)("+2) ^ n unh(1,Q)= ( , n-i)—2 . n ^ 0.

(-1,q )n+3 k=0

n > 0.

By virtue of the previous relation and the fact that 00^(1, q) = 1, we obtain relation (18). □

Theorem 2. We have the following connection formulas:

fn+2(x, q) = Ùn+2(x, q) - (1 + qn+i){1 + qn+2) l~Jn(x, Q), n > 0 (19)

Qn+1 _ 1 _

Hq(Tn+i(x, q)) = --— Un(x, q), n ^ 0, (20)

q — 1

(x2 - 1)Un(x, q) = Tn+2(x, q) + bnTn(x, q), n ^ 0, (21)

an+1 _ 1 ( „ „ (x2 - 1) Hg (Tn+1(x, q)) = q {Tn+2(x, q) + bnTn(x, q)j, n ^ 0, (22)

where

bo = -~q, bn = -(TTW1+qnTr), n ^ 1

Proof. Based on relation (11), we learn that [2]

Tn+2(x, q) = Un+2(x, q) + an(ln(x, q),n ^ 0,

where

Un+2(h q) - ^ffii(1, q) > 0 r . )

an =--=-1 ~m-, n ^ 0, U-i(1, q) = 0.

Un(1, q) - ^rUm^ q)

By virtue of Lemma 5, we obtain

_ 1 ßn = - (TT^^TT^^, n ^0

This provides (19).

We know that the sequence j -^HTn+\(x, g)| is a Hq-classical

orthogonal sequence with respect to Tq1 [6]. Moreover, by (9) and formula (2.9) in [6, p.58], we get

1 — n-3

Hq((x2 - q^Tf^ + LJL-XT^ = 0. Comparing the previous equation with the equation (14), we obtain

^ = V 2 ^.

Whence, Un(x, q) = q„+i\ HqTn+i(x, q), n > 0. Thus, we get (20).

qn+1-1 q

From the functional equation (11), we have [13]

(x2 - 1)Un(x, q) = Tn+2(x, q) + bnTn(x, q), n > 0, (23)

• ji 7 {Tq, (x2 - 1)Un(x, q)fn(x, q))

with bn =-t-, n > 0.

{Tq ,rT2(x, q))

Equivalently,

b = {(X - 1)Tq, Un(x, q)fn(x, q)) n > 0 n {Tq(x, q))

and, by the formula (11), we obtain bn — {'/Tq ^ ^

1 + q {Tq ,n (x, q)) n > 0. Therefore,

bn = - & ,in (X, Q)) ,n > 0. (24)

1 + Q {Tq,T2n(X, q)) , 1 '

On one hand, we have

{%,t2(x,q)) = tP{%,гг2-l(x,q)), n > 1

n

So, {Tq ,T2 (x, q)) = n 7fc9 ,n > 1, and, by (9), we obtain

k=i

n(n I

{tq ,T2 (x, q)) =4 (1 + qn)q ~, n > 1. (25)

+

k=0

(n (1

k=0

On the other hand, we may write

n

{LTq ,in (x, q)) = n^"9 ,n > 1. k=i

Using relation (10), we get

~ ~ (1+ o2+1)o

{Uq i (x, q)) = 4(1 + q)( + ^ , n > 0. (26)

/n+i

(n (1 + q

k=0

Relations (23), (24), (25), (26) and the fact that b0 = — , give (21).

After multiplying both sides of equation (20) by x2 — 1 and using relation (21), we deduce (22). □

Remark. When q ^ 1 in equations (19), (20), (21) and, (22) respectively, we meet, again, the formulas given in [10], [16], [17] concerning the classical monic Chebyshev polynomials.

4. Integral representation of Tq and Uq.

Theorem 3. For f <eV, we have

+-

{Uq, f) = Kf (x) q2)~ f(x) dx, 0 <q< 1, (27)

J (qx2; q2)^ -i

+q 2

{uq, f) = K J (p^. ^ f(x) dx,q> 1, (28)

i

-q22

_ 1 +q 2

_ i +q 2

+ u q^Ki [ x;f)™ dr

+ 2\ + q + 1 J (x — 1)(q2x2; q2)^

_ i

1 +q 2

+ ^ I (1 (qfj f 2) f(x) dx, 0 <q< 1, (29) q + 1 J (1 — x2)(q2x2; q2)^

№f)=2 {1—tK I -xv!^ * x\i (-1)+

+ 1 (1 + «2Kii_^_dx

+ 2l + q+1 J (x — 1)(q — -x2. q

--

i

— n 2

i

— n 2

i

— n 2

+1

q2K\ (x2;q 2)«,

+ —T n-2\( -i 2 f(x)dx, Q> 1 (30)

q+1 J (1 - x2)(q 1x2; q 2)x

1

where

i

+1 +q2

K )-1 = ! (f^^ dx, (K )-1 = f2 x2 ^ dx. (31)

J (qx2; q2)œ J (q-2x2; q-2)^

-1 i 1 -q 2

Proof. We need the following formula [11]:

(x - a)-l(x - a)w = w - (w)08a, a G C,w E V'. (32)

From the definition of the form Uq, we have

(Uq, f) = (u( 2 ), f(q- 2x)), feV.

By virtue of Proposition 4.3 in [15], we get

i

+q 2

(Uq, f) = K1 i (q-t;Î)(" f (q - 2x) dx, 0 <q< 1, (33) j (x ; q

i

-q 2

+q

(Uq, f) = K i ; q-l )" f (q- 2 x) dx,q> 1, (34)

j (q x ; q )<x -q

where K1 and K are normalization constants.

The change of variable t = q-2x in (33) and (34) gives relations (27) and (28), respectively.

Taking into account the functional equation (11), we may write

(x - 1)-1(x - 1)(x + 1)Tq = -^(x - 1)-\_ 1 Uq,

and, by formula (32) and the fact that ((x + 1)Tq)0 = 1, we get

(x + 1)% = 01 - x - 1)-1hq_iUq.

Always by (32), it follows that

1

% = 6-- + (x + 1)--5- — —(x + 1)--(x — 1)--hq_ 1 Uq. But (x + 1)--£- = 22(s- — <L-); then

% = 2(8-- + S-) — x + 1)--(x — 1)--hq_ 2 Uq.

This implies for f EV

{%, f) = 1(f(—1) + f(1)) — f). (35)

where

A(f) = {(x + 1)--(x — 1)--h 2Uq, f).

q 2

We may write

A(f) = {h _ 2 Uq, 6-0--f) =

q 2

1 /h j. 2 f(x) —x(f(1) — f(—1)) — f(—1) — f(1) \

= 2\1,-x2—1-A

Therefore,

K( f) 1 /jj 2f(q-2x) — (q-22x + 1)f(1) + (q-2x — 1)f(—1) \ (36)

A( f) = ~2\U*,--A (36)

When 0 < q < 1, we get, by (27), A( f) = \kI x

x 7 (x2; q2U 2f(q-2x) — (q-+ 1)f(1) + (q-— 1)f(—1)) x

J (qx2; q2)<x q--x2 — 1 '

-Let t = q-^x, it follows that

_ i +q 2

i

— n 2

+g 2

22

+ (J ît„ ^(-D*

2* ^J (t + 1)(q2t2; q2)

-g 2

_ l +g 2

+2.„2\

_ i -g 2

By virtue of the previous relation and (35), we deduce (29). In the case of q > 1, we get, by (28) and (36),

M( f) = 1K2 x

1

+g 2

x f (q-l%2; g-2U 2f(q-*x) - (q-2x + 1)f(1) + (q-2x - 1)f(-1) ^

J (q-2x2; q-2)<x q-lx2 - 1 '

i

-g 2

Using the change of variable y = q~2x, we obtain +l

l

(v<i~2U f (v)

* = {tJ 2 ! IV+

-1

+1

+ 22Kï(f ( +\Vq-lCO 2) dy) f(-1) +

2 V (y+1)(q V;q 2)^ )

-1

+1

+ (f ff^*)^

2 2 (1 - )( -1 2; -2) -1

Taking into account the last relation and (35), we get (30). □

Corollary 1. When q ^ 1 in representation (29) (respectively, (30)), we obtain the integral representation of T.

Proof. We need the following relations [2]:

+1 _ +1 +

. dx = . dx = n. (37)

1 + x 1 - x

-1 -1

Using relation (7), we obtain, successively:

Tim (x2: ft™ = lim (qf: ^ = VT—x2, |x| < 1. (38) q^i (qx2, q2)™ q^i (q2x2, q2)™

lim. (X ,q~\ = lim (q 'Y" q~2™ = VT-X, |x| < 1 (39) q^1 (q-ix2, q~2)™ q^1 (q~2x2, q-2)™

Based on relations (38) and (2), we get

+i

limKl = lm( (x) q2]™ dx)~l= --1-= 2. (40)

1 ç^AJ (qx2; q2)™ ) +l --k v 7

-i J V1 — x2 dx

i

On one hand, by (38) and (37) we have

_ i

+q 2 +i

„2. „2

(qx2; q2)™ _ f VT—

x

^ J (r+w&ïïzdx=J vmdx=k■ (41)

-1 -i -q 1 1

On the other hand, by relations (38) and (37), we obtain

_ i

+q 2 +i _

lim [ --(qX..; 2 2™ 2N dx = — [ 1 + x dx = —k. (42)

q^1 J (x — T)(q2x2; q2)™ J y/T—x v '

i -i -q - 2

Using relation (38), we see that

+q 2 +-

} / (qxq )<x> f(x) = [ f(x) (43)

%% J (1 —x2)(q2x2; q2)^dx = ] ^ 1 E' . (43)

Taking into account relations (38)-(43) and (30), we get

+11

\im{Tq, f) = - f (x) dx = {T, f), fEV.

q^- k J y/1 — x2

--

In a similar way, we obtain, from (39):

limK = -.

Also, by relations (39) and (38), we can deduce, successively,

+1 +1 _

lim [-(—————rrdx = [ , dx = n. (44)

(x +1)(q-1x2; q-] J Vx + 1

+1 +1 _

lim [ --(x.; q - dx = - [ }+Xdx. (45)

^J (x - 1)(q-1x2; q-2)<x J y/—x v '

-1 -1

By (39), we get

+1 +1

lim [ y,-(2xj;<11 2N f(x) dx = i J(X) dx, feV. (46)

q^1j (1 -x2)(q-1x2; q-2)<x J^ï-x2 ,J V J

-1 -1

Based on relations (44), (45), (46), and (31), we obtain

+1

11

\im(Tq ,f ) = - f (x)dx = (T, f), fEV.

v^1 n J y/1 - x2

-1

Hence, the desired results. □

In the following section, we give explicitly the expression of the Stielt-jes functions of the forms Tq and Uq.

5. The Stieltjes functions of the forms Tq and Uq.

Lemma 6. We have

(z2 - q2)Hq-i (S(z,%)) = -qzS(z,%), (47)

(z2 - q)Hq-i ( S(z, Uq )) = z S (z, Uq ) + q+1. (48)

Proof. We need the following formulas [2]:

w(1)(x) = 1, w(£)(x) = x, w eV' (symmetric form). (49)

From the functional equation in (9) and Theorem 2, we have

{q-2z2 — 1)Hg-i (S (z, %) = —(Hq-i (z2 — 1)) — z)( S (z,% )(z) — — Hq-i (Tq0o(x2 — 1))(z) — q(Tq0o)(—q--x)(z).

However, Hg-i(z2 — 1)) — z = — q--z, and with (49), we have

—Hq-i (%£ )(z) + (% 1)(z) = — Hq-i (Z) + (%1)(z) = 0.

Thus, we conclude the relation (47).

By the functional equation in (10) and Theorem 2, we may write

_2

{q-2z2 — q--)Hq-i (S (z, Uq )) = — (Vi (z2 — * )(S (z,% )(z) —

— Hg-i (Uq do (x2 — q--))(z) — q(Uq do)( x)(z),

since Hq-i(z2 — q--) + ^^z = — q-2z. By (47),

— Hq-i (Uqdo (x2 — q--))(z) — q(Uq do)C—^x)(z)

1—

1 — q-3

2

— Hq-i (x)(z) — q-—^Uq (1)(z) = q-- + q-2.

Which proves relation (48). □ Theorem 4. The following formulas hold:

S ex ) = — ^^ (50>

s ^ )=—1 • w >=0, (51)

S-u,) = a + ,->){^ (g-^t - 4 w <=0- (52>

s■>=(1+^- 4 w > ^=0 (53>

Proof. Equation (47) can be written as follows:

qz2_q2

s(q-lz, Tq) = —z-TS(z, Tq).

z2 - q2

Therefore, s(—, Tq) = q-1—s(1, Tq), z = 0. \qz / 1 - q2 z2 \ z J

Let

Ki, Tq) = ZA(Z), Z = 0. (54)

Then

1 - 2 1 - 2

This implies

A(qz) = ~-A(z), z = 0. (55)

A(z) = a (J^ , \q\ < 1, z = 0,a E C. (qz2, q2)™

By (54), we get

a (q2z-2, q2)

s (z, %) = -( VV ™ , \q\ < 1, z = 0.

z (qz 2, q2)™

( 2 2, 2)

But 1S(1, T) = a-—0 ' . ™ , and lim -S(1,Ta) = — 1. Then a = — 1, z Vz' q> (qz2, q2)™ z^oz Vz) q) '

which provides (50).

From relation (55), we get

.-1. 1 - ^

A(q-lz) = l 2 A(z), z = 0.

1 - -l

Thus,

A(z) = 3 ^t^™, z = 0, \v\ > 1,P E C, (z , q )™

and, by relation (54), it follows that

s, T.) = l -VA> ur. = 0.

Since lim -s(1 ,Tq) = -1, we obtain 3 = -1. So, we get (51).

z z

From the functional equation (11), we get

Uq = -(q + 1)h 2 ((x2 - 1)%), q2

and by formula (5), it follows that

Then

Uq = -(q + l)((q-1x2 - 1)hq 1 Tq).

S(z,Uq) = -(q + 1)5(z, (g-1 x2 - l)h i %))

q 2

and, by (4), we get S (z, Uq ) = -(q + 1)(q-1z2 - 1)S (z, h 2 Tq))-

q2

But

- (q + 1)(K 1 Tq)do(q-1x2 - 1))(z).

(h i Tq)90(q-1x2 - 1))(z) = q-1(h i Tq)(0(x) = q-1x,

S(Z,h 2Tq)) = q-2S(q-2z,Tq). q 2

Then

S(z,Uq) = -(q-1 + 1){q2(q-1z2 - 1)S(q-1 z,Tq) + z)). Using relations (50) and (51), we obtain formulas (52) and (53). □

Acknowledgments. I am very grateful to the editors and the referee for the constructive and valuable comments and recommendations.

References

[1] Atakishiyeva M, Atakishiyev N. On discrete q-extension of Chebyshev polynomials. Commun. Math. Anal., 2013, vol. 14(2), pp. 1-12.

[2] Ben Hadj ali B , Mejri M. Algebraic equation and symmetric second degree forms of class two. Integral Transforms Spec. Funct., 2017, vol. 28(9), pp. 682-701. DOI: https://doi.org/10.1080/10652469.2017.1345903

[3] Chihara T. S. An introduction to orthogonal polynomials. Gordon and Breach, New york, 1978.

[4] Ecran E, Cetin M. Tuglu N. Incomplete q-Chebyshev polynomials. Filo-mat., 2018, vol. 32(10), pp. 3599-3607.

DOI: https://doi .org/10.2298/FIL1810599E

[5] Gasper G, Rahman M. Basic Hypergeometric Functions. 2nd ed. Cambridge Univ. Press 2004.

DOI: https://doi .org/10.1017/CBCI9780511526251

[6] Kheriji L , Maroni P. The Hq-classical orthogonal polynomials. Acta. Appl. Math., 2002, vol. 71, pp. 49-115.

DOI: https://doi.org/10.1023/A:1014597619994

[7] Kheriji L. An introduction to the Hq-semiclassical. Methods Appl. Anal., 2003, vol. 10(3), pp. 387-412.

[8] Kizilates C, Tuglu N, Cekim B. On the (p,q)-Chebyshev polynomials and related polynomials. Mathematics, 2019, vol. 7(136), pp. 1-12.

DOI: https://doi .org/10.3390/math7020136

[9] Koekoek R, Lesky P. A, Swarttouw R. F. Hypergeometric Orthogonal Polynomials and Their q-Analoguess. Springer. 2010.

DOI: https://doi.org/10.1007/978-3-642-05014-5

[10] Maroni P. An introduction to second degree forms. Adv Comput Math., 1995, vol. 3(1-2), pp. 59-88.

DOI: https://doi .org/10.1007/BF02431996

[11] Maroni P. Une theorie algebrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and their applications. Proc. Erice (1990), IMACS, Ann. Comput. Appl. Math., 1991, vol. 9, pp. 95-130.

[12] Maroni P. Fonctions eulriennes. Polynmes orthogonaux classiques. A54v1, 1994, pp. 1-30.

[13] Maroni P. Semi-classical character and finite-type relations between polynomial sequences. Appl Numer Math., 1999, vol. 31(3), pp. 295-330. DOI: https://doi.org/10.1016/S0168-9274(98)00137-8

[14] Maroni P, Mejri M. The I(q,w) classical orthogonal polynomials. Appl. Numer. Math., 2002, vol. 43(4), pp. 423-458.

DOI: https://doi.org/10.1016/S0168-9274(01)00180-5

[15] Mejri M. q-Extension of generalized Gegenbauer polynomials. J. Difference Equ. Appl., 2010, vol. 16(12), pp. 1367-1380.

DOI: https://doi.org/10.1080/10236190902821671

[16] Rivlin T. J. The Chebyshev Polynomials. Wiley-Interscience, New York, 1974.

[17] Spencer B. G. The classical orthogonal polynomials. World Scientific Publishing CO Pte Ltd, 2016.

Received May 22, 2021. In revised form,, October 12, 2021. Accepted October 15, 2021. Published online December 5, 2021.

University Tunis El Manar, Preparatory Institute for Engineering Studies, Campus universitaire El Manar-B.P 244, El Manar II, Tunis 2092, Tunisia E-mail: [email protected]