Download - -orthogonality of discrete -Hermite type polynomials

Available online at www.sciencedirect.com

Journal of Approximation Theory 170 (2013) 116–133www.elsevier.com/locate/jat

Full length article

d-orthogonality of discrete q-Hermite type polynomials

Imed Lamiri

Faculte des Sciences de Monastir, Departement de Mathematiques, 5019 Monastir, Tunisia

Available online 1 August 2012

Communicated by Walter Van Assche

Abstract

In this paper, we solve a characterization problem involving a suitable basic-hypergeometric form ofa polynomial set. That allows us to introduce new examples of Lq -classical d-orthogonal polynomials,generalizing the discrete q-Hermite polynomials in the context of d-orthogonality, and a q-analogous forthe d-orthogonal polynomials of Gould–Hopper. For the resulting polynomials, we derive miscellaneousproperties. Those turn out to be limit relations, recurrence relations of order (d + 1), difference formulas,generating functions, inversion formulas, and d-dimensional functional vectors.c⃝ 2012 Elsevier Inc. All rights reserved.

Keywords: d-orthogonality; Basic hypergeometric polynomials; Linear functionals; Hermite polynomials; Discreteq-Hermite polynomials I and II

1. Introduction

Let P be the vector space of polynomials with coefficients in C and let P ′ be its algebraicdual. We denote by ⟨u, f ⟩ the effect of the functional u ∈ P ′ on the polynomial f ∈ P . Apolynomial sequence {Pn}n≥0 is called a polynomial set (PS, in short) if and only if deg Pn = nfor all non-negative integer n.

Definition 1.1 (Van Iseghem [34] and Maroni [31]). Let d be a positive integer. A PS {Pn}n≥0is called d-orthogonal (d-OPS, in short) with respect to the d-dimensional vector of functionalsΓ =

t (Γ0,Γ1, . . . ,Γd−1) if it satisfies the following orthogonality relations:

E-mail address: [email protected].

0021-9045/$ - see front matter c⃝ 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jat.2012.07.002

http://www.elsevier.com/locate/jat

http://dx.doi.org/10.1016/j.jat.2012.07.002

http://www.elsevier.com/locate/jat

mailto:[email protected]

http://dx.doi.org/10.1016/j.jat.2012.07.002

I. Lamiri / Journal of Approximation Theory 170 (2013) 116–133 117⟨Γk, Pr Pn⟩ = 0, r > nd + k, n ∈ N = {0, 1, 2, . . .},

⟨Γk, Pn Pnd+k⟩ = 0, n ∈ N,

for each integer k belonging to Nd = {0, 1, . . . , d − 1}.

For d = 1,we have the well-known notion of orthogonality.Many examples of d-OPSs were introduced by solving various characterization problems

which consist of finding all d-orthogonal polynomials satisfying a fixed property (see, for in-stance, [5,11,28,12,6–9,13–15,10,16,19–21,31,22,23,29,36]). In particular, in [29], we solve thefollowing characterization problem:

P1. Find all d-orthogonal polynomials of the form:

Pn(x; c, m, (ap), (bq)) = xnm+p Fq

∆(m, −n), (ap)

(bq);

m

cx

m

, (1.1)

where• (ap) designates the set {a1, a2, . . . , ap},

• (a)p is the Pochhammer symbol defined by (a)p =Γ (a+p)Γ (a)

,

• [ar ]p denotes the product [ar ]p =r

i=1(ai )p,

• ∆(r, α) indicates the array of r parameters α+ j−1r , j = 1, . . . , r ,

• p Fq(z) denotes the generalized hypergeometric function with p numerator and q denom-inator parameters defined by Luke [30]

p Fq

(ap)

(bq); z

:=

∞k=0

[ap]k

[bq ]k

zk

k!. (1.2)

We obtain that the Gould–Hopper generalization of the Hermite polynomials [24, p. 58]:

gd+1n (x; h) = xn

d+1 F0

∆(d + 1, −n)

−; h

−(d + 1)

x

d+1

, (1.3)

is the only solution of the Problem P1. These polynomials appear among the solutions oftwo other characterization problems arising in the d-orthogonal polynomials theory andconsidered by Ben Cheikh and Douak [6], and the author and Ouni [29]. Furthermore,many properties of these polynomials are singled out in [6,29], we quote for instance:generating function, inversion formula, recurrence relation, differential equation, andfunctional ensuring their d-orthogonality. So it is of interest to consider the followingq-analogous of Problem P1:

P2. Find all d-orthogonal polynomials of type:

Pn(x; (ap), (bl)|q)

= xnm+pΦl

q−n, q−n+1, . . . , q−n+m−1, (ap)

(bl)

qm;

q(sn+1)m

cm xm

, (1.4)

where m ≥ 2, 0 < q < 1, c ∈ C, s ∈ N := {0, 1, 2, . . .} and ai ; i = 1, . . . , p and b j ; j =

1, . . . , l are p + l complex numbers such that: ai = 0; ai = b j and ai , b j = 1, q−1, q−2, . . . rφsdesignate the q-hypergeometric series defined by Koekoek and Swarttouw [27, p. 11]:

rφs

a1, . . . , arb1, . . . , bs

q; z

118 I. Lamiri / Journal of Approximation Theory 170 (2013) 116–133

:=

∞n=0

(a1, . . . , ar ; q)n

(b1, . . . , bs; q)n(−1)n(1+s−r)q( n

2 )(1+s−r) zn

(q; q)n, (1.5)

with (a1, . . . , ar ; q)n :=r

j=1(a j ; q)n, r a positive integer or 0 (interpreting an empty product

as 1), (a; q)0 := 1, (a; q)n :=n−1

j=0(1 − aq j ), n > 1, and (a; q)∞ := limn →+∞(a; q)n .

Such a characterization takes into account the fact that PS which are obtainable from oneanother by a linear change of variable are assumed equivalent.

The polynomials given by (1.4) contain as particular cases the discrete q-Hermite Ipolynomials (m = 2, s = 1, p = l = 0, c2

= q3) given by Koekoek and Swarttouw[27, p. 118]:

hn(x; q) = xn2Φ0

q−n, q−n+1,

−

q2;

q2n−1

x2

, (1.6)

and the discrete q-Hermite II polynomials (m = 2, s = p = 0, l = 1, b1 = 0, c2= −1) given

by Koekoek and Swarttouw [27, p. 119]:

hn(x; q) = xn2Φ1

q−n, q−n+1,

0

q2; −

q2

x2

. (1.7)

As a solution of Problem P2 we obtain as far as we know two new d-OPSs which represent, onthe one hand, a new generalization of the discrete q-Hermite polynomials I and II in the contextof d-orthogonality, and on the other hand, the q-analogous of the Gould–Hopper polynomials.

For the resulting polynomials, we derive miscellaneous properties. Those turn out to be: limitrelations, recurrence relations of order (d + 1), difference formulas, generating functions, inver-sion formulas, and d-dimensional functional vectors.

The paper is organized as follows: Section 2, is devoted to solve the characterization ProblemP2. Section 3 deals with some properties of the obtained polynomials.

2. d-orthogonality of discrete q-Hermite type polynomials

In this section, to solve Problem P2, we state the following characterization theorem.

Theorem 2.1. The only d-orthogonal polynomials of type (1.4) arise in the following cases.

Case 1. m = d + 1, s = 1, p = l = 0.

Case 2. m = d + 1, s = p = 0, l = m − 1, b j = 0; j = 1, . . . , l.Moreover, the PSs given by (1.4) in these two cases verify the recurrence relation:

x Pn(x; (ap), (bl)|q) = Pn+1(x; (ap), (bl)|q) + α(s)n Pn+1−m(x; (ap), (bl)|q), (2.1)

where

α(s)n =

(−1)1+l−m

cm q(n+1)(s−1)+m(sn+1)(q−n; q)m−1. (2.2)

To prove this theorem we need the following definition and two lemmas.

I. Lamiri / Journal of Approximation Theory 170 (2013) 116–133 119

Definition 2.2 ([32]). A PS {Pn}n≥0 is called d-symmetric if it fulfills

Pn(ωd x) = (ωd)n Pn(x), n ≥ 0, ωd = exp

2iπ

d + 1

.

Lemma 2.3 ([31]). A PS {Pn}n≥0 is d-orthogonal if and only if it satisfies a (d + 1)-orderrecurrence relation of the type:

x Pn(x) =

d+1k=0

αk,d(n)Pn−d+k(x), (2.3)

where αd+1,d(n)α0,d(n) = 0, n ≥ d, and by convention, P−n = 0, n ≥ 1.

Lemma 2.4 ([28]). Let {Pn}n≥0 be a (m − 1)-symmetric PS defined by:

Pn(x) =

[nm ]

k=0

γn,m(k)xn−mk .

If {Pn}n≥0 is a d-OPS, then there exists a positive integer r0 and a sequence {α(r0−r)m+1(n)}0≤r≤r0≤[

nm ] such that: d = r0m − 1, α1(n)αr0m+1(n) = 0 and

min(k,r0)r=0

α(r0−r)m+1(n)γn+1−rm,m(k − r) = γn,m(k); n ≥ r0m; k = 0, . . . , n

m

.

Proof of Theorem 2.1. This proof is divided into two steps. The first one is devoted to showingthat if {Pn}n≥0 is a d-OPS of the form (1.4) then we have one of the two cases cited inTheorem 2.1. In the second step, we prove the converse and the recurrence relation (2.1).

Let {Pn}n≥0 be a PS of the form (1.4). Using (1.5), (1.4) and the identity:

m−1j=0

(q−n+ j; qm)k = (q−n

; q)km, (2.4)

we have

Pn(x; (ap), (bl)|q) =

[nm ]

k=0

γn,m(k)xn−mk, (2.5)

where

γn,m(k) =(q−n

; q)mk

(qm; qm)k

pj=1

(a j ; qm)k

lj=1

(b j ; qm)k

(−1)k(1+l−p−m)

cmk qm

k2

(1+l−p−m)+(sn+1)mk

. (2.6)

According to Lemma 2.4, we deduce that there exist a positive integer r0 and a sequence{α(r0−r)m,d(n)}0≤r≤r0≤[

nm ] such that: d = r0m − 1, α0,d(n)αr0m,d(n) = 0 and

min(k,r0)r=0

α(r0−r)m,d(n)γn+1−rm,m(k − r) = γn,m(k); n ≥ r0m; k = 0, . . . , n

m

. (2.7)


In particular, for k = 0, we get:

αr0m,d(n)γn+1,m(0) = γn,m(0).

However, γn,m(0) = 1. Then αr0m,d(n) = 1.On the other hand, since r0 ∈ {0, . . . , [ n

m ]} and k is an arbitrary integer in {0, . . . , [ nm ]}, we

can assume that k ≥ r0. Substituting (2.6) in (2.7) and using the identity:

(a; qm)k−r =(a; qm)k

(aqm(k−r); qm)r,

we obtain

1 =

r0r=0

ξr (n)(qm(1+k−r)

; qm)r

(1 − q−n−1+mk)

lj=1

(b j qm(k−r); qm)r

pj=1

(a j qm(k−r); qm)r

qmk[s(1−rm)−r(1+l−p−m)], (2.8)

where

ξr (n) = α(r0−r)m,d(n)(1 − q−n−1)

(q−n−1; q)rm(−1)r(1+l−p−m)

× cmr qm

r+12

(1+l−p−m)−mr(sn+1)−srm(1−rm)

. (2.9)

Let A and B be two polynomials such that:

A

B(x) =

r0r=0

ξr (n)(qm(1−r)x; qm)r

(1 − q−n−1x)

lj=1

(b j q−rm x; qm)r

pj=1

(a j q−rm x; qm)r

x s(1−rm)−r(1+l−p−m),

x ∈ C. (2.10)

According to (2.8), we deduce that: A(x) = B(x), for x = qmk , k = 0, . . . , [ nm ], and n ∈ N.

Hence:A

B(x) = 1, ∀x ∈ C.

By using (2.10) and the fact that αr0m,d(n) = 1, we obtain:

(1 − q−n−1x) = (1 − q−n−1)x s+

r0r=1

ξr (n)Ar (x), (2.11)

where

Ar (x) = (qm(1−r)x; qm)r

lj=1

(b j q−rm x; qm)r

pj=1

(a j q−rm x; qm)r

x s(1−rm)−r(1+l−p−m), r = 1, . . . , r0.

Let l1 be the number of b j ; j = 1, . . . , l; satisfying b j = 0, and let l2 = l − l1. Then we have:

Ar (x) ∼ x s+r [m(1−s)−l2], x ↑ ∞, r = 1, . . . , r0, (2.12)


where

f (x) ∼ g(x), x ↑ ∞ ⇔ limx−→∞

f (x)

g(x)= 1.

Suppose that: s > 1. By using (2.11) and (2.12), we deduce that there exists a positive integer r1in {1, . . . , r0} such that s = s +r1[m(1− s)− l2]. However m(1− s)− l2 < 0. Therefore r1 = 0,which is impossible since r1 ≥ 1. Consequently s ≤ 1.

Now, we need to consider two cases.

Case 1. s = 1. In this case (2.12) becomes: Ar (x) ∼ x (1−rl2), x ↑ ∞, r = 1, . . . , r0.If we suppose that l2 > 0, we get

limx→∞

Ar (x) = 0 or 1, ∀r ∈ {1, . . . , r0}.

It follows from (2.11) that

(1 − q−n−1x) ∼ (1 − q−n−1)x, x ↑ ∞,

which is impossible. Therefore l2 = 0.Now, replacing s by 1 and l2 by 0 in (2.11) to obtain:

(1 − q−n−1x) = (1 − q−n−1)x

+

r0r=1

ξr (n)(qm(1−r)x; qm)r

l1j=1

(b j q−rm x; qm)r

pj=1

(a j q−rm x; qm)r

x1−r(1+l1−p). (2.13)

By letting x = 0 in this identity, we deduce that 1 − r(1 + l1 − p) ≥ 0; r = 1, . . . , r0; and thereexists a positive integer r2 in {1, . . . , r0} such that 1 − r2(1 + l1 − p) = 0. Hence r2 = r0 = 1and l1 = p. That implies m = d + 1, and (2.13) becomes:

ξ1(n)

l1j=1

(1 − b j q−m x)

(1 − a j q−m x)= 1,

which is impossible, since ai = b j for i, j = 1, . . . , l1. Then l1 = 0. So, in this case, we obtain:s = 1, l = p = 0 and m = d + 1. That corresponds to the first case cited in Theorem 2.1.

Case 2. s = 0. In this case (2.12) becomes: Ar (x) ∼ xr(m−l2), x ↑ ∞, r = 1, . . . , r0.Using (2.11), we deduce that there exists a positive integer r3 in {1, . . . , r0} such that

r3(m − l2) = 1. Hence r3 = 1 and l2 = m −1, which imply Ar (x) ∼ xr , x ↑ ∞, r = 1, . . . , r0.That, by virtue of (2.11) leads to r0 = 1. Then m = d + 1 and (2.11) becomes:

ξ1(n)qn+1x p−l1

lj=1

(1 − b j q−m x)

pj=1

(1 − a j q−m x)r

= 1.

By letting x = 0 in this identity, we deduce p = l1. Furthermore, from the fact thatlj=1(1 − b j q−m x) and

pj=1(1 − a j q−m x) are coprime, we deduce p = l1 = 0. Then, in

this case we get: s = 0, m = d + 1, p = 0, l = m − 1 and b j = 0; j = 1, . . . , l. Thatcorresponds to the second case of Theorem 2.1.


Next, we prove the converse and the recurrence relation (2.1).For the polynomials given by (1.4) in cases 1 and 2 of Theorem 2.1, the identity (2.6) is

reduced to the following

γn,m(k) =(q−n

; q)mk

(qm; qm)kq

m

k2

(1+l−m)+(sn+1)mk (−1)k(1+l−m)

cmk . (2.14)

Therefore,

γn,m(k) − γn+1,m(k)

= qm

k2

(1+l−m)+(sn+1)mk (−1)k(1+l−m)

cmk(qm; qm)k[(q−n

; q)mk − qsmk(q−n−1; q)mk].

That by virtue of the following identities:

(q−n; q)mk = (1 − q−n−1+mk)(q−n

; q)m−1(q−n−1+m

; q)m(k−1)

(q−n−1; q)mk = (1 − q−n−1)(q−n

; q)m−1(q−n−1+m

; q)m(k−1),

leads to:


= qm

k2

(1+l−m)+(sn+1)mk (−1)k(1+l−m)

cmk(qm; qm)k(q−n

; q)m−1

× (q−n−1+m; q)m(k−1)[1 − q−n−1+mk

− qsmk(1 − q−n−1)]. (2.15)

On the other hand, it is obvious that:

(qm; qm)k = (1 − qmk)(qm

; qm)k−1,

[1 − q−n−1+mk− qsmk(1 − q−n−1)] = q(n+1)(s−1)(1 − qmk), s = 0, 1.

Substituting these identities in (2.15) and using (2.14), we obtain:


=(−1)(1+l−m)

cm q(n+1)(s−1)(q−n; q)m−1qm(k−1)[(m−1)(s−1)+l]+m(sn+1)

× γn+1−m,m(k − 1). (2.16)

However, [(m − 1)(s − 1) + l] = 0 for s = 0, 1. Then (2.16) becomes:

γn,m(k) − γn+1,m(k) = α(s)n γn+1−m,m(k − 1), (2.17)

where α(s)n is given by (2.2).

On the other hand, from (2.5), we have

Pn(x; (ap), (bl)|q) =

[n+1

m ]k=0

γn,m(k)xn−mk, (2.18)

since, [nm ] = [

n+1m ] or [

nm ] = [

n+1m ] and γn,m([ n+1

m ]) = 0. Hence, by using (2.17), we can write:

x Pn(x; (ap), (bl)|q) = xn+1+

[n+1

m ]k=1

γn,m(k)xn+1−mk,


= xn+1+

[n+1

m ]k=1

[γn+1,m(k) + α(s)n γn+1−m,m(k − 1)]xn+1−mk,

=

[n+1

m ]k=0

γn+1,m(k)xn+1−mk+ α(s)

n

[n+1−m

m ]k=0

γn+1−m,m(k)xn+1−m−mk,

= Pn+1(x; (ap), (bl)|q) + α(s)n Pn+1−m(x; (ap), (bl)|q).

Then, the PS {Pn}n≥0 verifies the recurrence relation (2.1). Furthermore, from Lemma 2.3, wededuce that {Pn}n≥0 is d-orthogonal. �

Remark 2.5. If we take cm= qm+1 in the first case of Theorem 2.1, and cm

= −1 in the secondcase of this theorem, we obtain the following classes of q-polynomials:

P(1)n (x |q) = xn

d+1Φ0

q−n, q−n+1, . . . , q−n+d

−

qd+1;

qn(d+1)−1

xd+1

, (2.19)

P(2)n (x |q) = xn

d+1Φd

q−n, q−n+1, . . . , q−n+d

0, . . . , 0

qd+1;−q(d+1)

xd+1

. (2.20)

These PSs appear to be new d-OPSs for d = 1. The PS given by (2.19) (respectively (2.20)) willbe called discrete q-Hermite I type PS (respectively, discrete q-Hermite II type PS), since, ford = 1, they are reduced to the discrete q-Hermite I and II PS respectively, and according to theproperties obtained in Section 3, which are analogous to the discrete q-Hermite ones.

For d = 1, Theorem 2.1 is reduced to the following.

Corollary 2.6. A PS {Pn}n≥0 defined by (1.4) with m = 2 is orthogonal if and only if it is thediscrete q-Hermite I or II PS given by (1.6) and (1.7) respectively.

3. Some properties of the discrete q-Hermite type polynomials

In this section we focused our interest to derive some properties of the discrete q-Hermitetype d-OPSs, generalizing in a natural way the discrete q-Hermite ones.

3.1. Link between the discrete q-Hermite type polynomials I and II

Proposition 3.1. The discrete q-Hermite I type polynomials given by (2.19) are related to thediscrete q-Hermite II type polynomials given by (2.20) in the following way:

P(1)n

ei π

d+1 q−(d−1)(d+2)

2(d+1) x |q−1

= ei nπd+1 q

−n(d−1)(d+2)2(d+1) P(2)

n (x |q). (3.1)

Proof. From the identity (2.19), we have:

P(1)n (x | q−1) =

[n

d+1 ]k=0

dj=0

(qn− j; q−d−1)k

(q−d−1; q−d−1)k(−1)dkq

d(d+1)

k2

−k(n(d+1)−1)

xn−(d+1)k .


That by virtue of the transformations:

(a; q−1)k = (a−1; q)k(−a)kq

−

k2

, a = 0, (3.2)

(a; q)(d+1)k =

dj=0

(aq j; qd+1)k, (3.3)

leads to

P(1)n (x | q−1) =

[n

d+1 ]k=0

(q−n; q)(d+1)k

(qd+1; qd+1)k(−1)kq(d+1)k

e−i π

d+1 q(d−1)(d+2)

2(d+1)

−k(d+1)

xn−(d+1)k .

In this identity replacing x by (ei πd+1 q

−(d−1)(d+2)2(d+1) )x to obtain (3.1). �

Remark 3.2. For d = 1, the equality (3.1) is reduced to the known relationship [27, p. 118]:

hn(i x |q−1) = in hn(x |q),

involving the discrete q-Hermite I polynomials defined by (1.6) and the discrete q-Hermite IIpolynomials given by (1.7).

3.2. Limit relations

Proposition 3.3. The Gould–Hopper polynomials defined by (1.3) can be obtained from the dis-crete q-Hermite I type polynomials given by (2.19) in the following way:

limq→1

(1 − qd+1)−dn

d+1 P(1)n ((1 − qd+1)

dd+1 x |q) = gd+1

n

x;

−1

(d + 1)d+1

. (3.4)

The Gould–Hopper polynomials can also be found from the discrete q-Hermite II type polyno-mials given by (2.20) in a similar way:

limq→1

(1 − qd+1)−dn

d+1 P(2)n ((1 − qd+1)

dd+1 x |q) = gd+1

n

x;

(−1)d

(d + 1)d+1

. (3.5)

Proof. Let Pn(x |q) be the discrete q-Hermite type polynomials I and II. Starting from (2.19) and(2.20), we have:

Pn(x |q) =

[n

d+1 ]k=0

γn,d+1(k)xn−(d+1)k,

where

γn,d+1(k) =

dj=0

(q−n+ j; qd+1)k

(qd+1; qd+1)kq

(d+1)

k2

(l−d)+(sn+1)(d+1)k (−1)k(l−d)

c(d+1)k, (3.6)

with s = 1, l = 0, cd+1= qd+2 for the discrete q-Hermite I type polynomials, and s = 0,

l = d, cd+1= −1 for the discrete q-Hermite II type polynomials.


Therefore,

(1 − qd+1)−dn

d+1 Pn((1 − qd+1)d

d+1 x |q) =

[n

d+1 ]k=0

(1 − qd+1)−dkγn,d+1(k)xn−(d+1)k . (3.7)

On the other hand, the identity (3.6) leads to:

(1 − qd+1)−dkγn,d+1(k)

=

dj=0

(q−n+ j;qd+1)k

(1−qd+1)k

(qd+1;qd+1)k(1−qd+1)k

q(d+1)

k2

(l−d)+(sn+1)(d+1)k (−1)k(l−d)

c(d+1)k. (3.8)

Using the transformation formula [27, p. 7]:

limq→1

(qα; q)k

(1 − q)k = (α)k,

the equality (3.8) gives

limq→1

(1 − qd+1)−dkγn,d+1(k) =

dj=0

(−n+ jd+1 )k

k!(−1)(l−d)kτ−k, (3.9)

where τ = limq→1 cd+1= 1 for the discrete q-Hermite I type polynomials, and τ = limq→1

cd+1= −1 for the discrete q-Hermite II type polynomials.

From (3.7) and (3.9), we deduce:

limq→1

(1 − qd+1)−dn

d+1 Pn((1 − qd+1)d

d+1 x |q) = xnd+1 F0

∆ (d + 1, −n)

−;(−1)(l−d)

τ xd+1

.

The desired result follows from the last equality and the identity (1.3). �

Remark 3.4. 1. For d = 1, the identities (3.4) and (3.5) are reduced, respectively, to the knownlimit relations [27, p. 144]:

limq→1

hn(x

1 − q2 | q)

(1 − q2)n2

=Hn(x)

2n , (3.10)

limq→1

hn(x

1 − q2 | q)

(1 − q2)n2

=Hn(x)

2n , (3.11)

involving the classical Hermite polynomials Hn , the discrete q-Hermite I polynomials hngiven by (1.6) and the discrete q-Hermite II polynomials hn given by (1.7).

2. The limit relations given by (3.4) and (3.5) prove that the obtained discrete q-Hermite typepolynomials I and II represent two q-analogs of the Gould–Hopper polynomials.

3.3. Recurrence relations

From Theorem 2.1, we have the following.

Proposition 3.5. The discrete q-Hermite I type polynomials given by (2.19) verifies the followingrecurrence relation:


x P(1)n (x |q) = P(1)

n+1(x |q) + (−1)dqn(d+1)−1(q−n; q)d P(1)

n−d(x |q). (3.12)

The discrete q-Hermite II type polynomials given by (2.20) satisfy the following recurrencerelation:

x P(2)n (x |q) = P(2)

n+1(x |q) − q−n+d(q−n; q)d P(2)

n−d(x |q). (3.13)

Remark 3.6. For d = 1, the identities (3.12) and (3.13) are reduced, respectively, to the knownrecurrence relations [27, pp. 118–119]:

xhn(x |q) = hn+1(x |q) + qn−1(1 − qn)hn−1(x |q),

xhn(x |q) = hn+1(x |q) + q−2n+1(1 − qn)hn−1(x |q),

associated to the discrete q-Hermite I and II polynomials.

3.4. Difference formulas

Let q be a real number, Hahn [25] defined a linear operator Lq by

Lq( f )(x) =f (qx) − f (x)

(q − 1)x, |q| = 1, (3.14)

where f is a suitable function for which the second member of this equality exists. This operatortends to the derivative operator D as q −→ 1.

Definition 3.7. Let {Pn}n≥0 be a d-OPS. Put Qn(x) = Lq Pn+1(x), n ≥ 0. If the sequence{Qn}n≥0 is also d-orthogonal, the sequence {Pn}n≥0 is called Lq -classical d-OPS.

For d = 1, we meet the notion of the classical property according to Hahn’s property.Using this definition we have the following.

Proposition 3.8. The discrete q-Hermite I and II type polynomials given by (2.19) and (2.20) areLq -classical d-orthogonal polynomials. Moreover they satisfy:

Lkq P(1)

n (x |q) =(qn−k+1

; q)k

(1 − q)k P(1)n−k(x |q), k ≥ 1, (3.15)

Lkq P(2)

n (x |q) = q

k+1

2

(q−n

; q)k

(q − 1)k P(2)n−k(q

k x |q), k ≥ 1. (3.16)

Proof. Let Pn(x |q) be the discrete q-Hermite type polynomials I and II. Then, we have:

Pn(x |q) =

[n

d+1 ]k=0

γn,d+1(k)xn−(d+1)k,

where γn,d+1(k) is given by (3.6).According to (3.6), (3.3) and the following identities:

(q−n; q)(d+1)k =

1 − q−n

1 − q−n+(d+1)k(q−n+1

; q)(d+1)k,

Lq [xn−(d+1)k] =

1 − qn−(d+1)k

1 − qxn−1−(d+1)k,


we obtain

Lq Pn(x |q) =1 − qn

1 − q

[n−1d+1 ]k=0

(q−n+1; q)(d+1)k

(qd+1; qd+1)kq

(d+1)

k2

(l−d)+sn(d+1)k

×(−1)k(l−d)

c(d+1)kxn−1−(d+1)k . (3.17)

For the discrete q-Hermite I type polynomials (s = 1, p = l = 0, cd+1= qd+2), (3.17)

becomes

Lq P(1)n (x |q) =

1 − qn

1 − qP(1)

n−1(x |q). (3.18)

For the discrete q-Hermite II type polynomials (s = p = 0, l = d, cd+1= −1), (3.17) leads to

Lq P(2)n (x |q) = q−n+1 1 − qn

1 − qP(2)

n−1(qx |q). (3.19)

Hence the sequences {P(1)n (x |q)}n≥0 and {P(1)

n (x |q)}n≥0 are also d-orthogonal. So, according toDefinition 3.7, these d-OPSs are Lq -classical.

Now, the iteration of (3.18) and (3.19) leads to (3.15) and (3.16) respectively. �

Remark 3.9. For d = k = 1, the identities (3.15) and (3.16) are reduced to the known differenceformulas [27, pp. 118–119]:

Lq hn(x |q) =1 − qn

1 − qhn−1(x |q),

Lq hn(x |q) = q−n+1 1 − qn

1 − qhn−1(qx |q).

Proposition 3.10. The discrete q-Hermite I type polynomials given by (2.19) satisfy the follow-ing difference equation:

(−1)dq(n−1)(d+1)−1 (q−n+1; q)d(1 − q)d+1

(qn−d; q)d+1Ld+1

q y(x)

−1 − q

1 − qn x Lq y(x) + y(x) = 0, y(x) = P(1)n (x |q). (3.20)

Proof. According to (3.12), we have:

x P(1)n−1(x |q) = P(1)

n (x |q) + (−1)dq(n−1)(d+1)−1(q−n+1; q)d P(1)

n−1−d(x |q). (3.21)

On the other hand, using (3.15), we obtain:

P(1)n−d−1(x |q) =

(1 − q)d+1

(qn−d; q)d+1Ld+1

q P(1)n (x |q), (3.22)

P(1)n−1(x |q) =

1 − q

1 − qn Lq P(1)n (x |q). (3.23)

Now, substituting (3.22) and (3.23) in (3.21) to obtain (3.20). �


Remark 3.11. In the case when d = 1, the identity (3.20) is reduced to the following:

qn−2(1 − q)L2q [y(x)] − x Lq [y(x)] +

1 − qn

1 − qy(x) = 0, y(x) = hn(x | q). (3.24)

On the other hand, from (3.14), we have:

Lq [y](q−1x) =y(x) − y(q−1x)

(1 − q−1)x,

L2q [y](q−1x) =

y(qx) − (1 + q)y(x) + qy(q−1x)

q−1(1 − q)2x2 .

By substituting these identities in (3.24), we rediscover the difference equation associated to thediscrete q-Hermite polynomials [27, p. 118]:

y(qx) + [q−n+1x2− (1 + q)]y(x) + q(1 − x2)y(q−1x) = 0, y(x) = hn(x | q).

3.5. Generating functions

Proposition 3.12. The discrete q-Hermite I type polynomials given by (2.19) have the followinggenerating function:

q(d−1)(d+2)

2 td+1; qd+1

∞

(xt; q)∞=

∞n=0

P(1)n (x |q)

(q; q)ntn, (3.25)

where

1(t; q)∞

=

∞n=0

tn

(q; q)nand (−t; q)∞ =

∞n=0

q( n2 )

(q; q)ntn . (3.26)

The discrete q-Hermite II type polynomials given by (2.20) verify the following generatingrelation:

(−xt; q)∞

(−(−t)d+1; qd+1)∞=

∞n=0

q( n2 )

(q; q)nP(2)

n (x |q)tn . (3.27)

Proof. According to (2.19), we have:n≥0

P(1)n (x |q)

(q; q)ntn

=

n≥0

[n

d+1 ]k=0

(q−n; q)(d+1)k

(qd+1; qd+1)kq

−d(d+1)

k2

+[n(d+1)−1]k

(−1)dk xn−(d+1)k tn

(q; q)n

=

∞n,k=0

(q−(n+(d+1)k); q)(d+1)k

(q; q)n+(d+1)k(qd+1; qd+1)k

× q−d(d+1)

k2

+[(d+1)(n+(d+1)k)−1]k

(−1)dk xn tn+(d+1)k .


On the other hand, it is obvious that:

(q−[n+(d+1)k]; q)(d+1)k

(q; q)n+(d+1)k=

(−1)(d+1)k

(q; q)nq

(d+1)k

2

−[n+(d+1)k](d+1)k

,

k = 0, . . . ,

n

d + 1

consequently

n≥0

P(1)n (x |q)

(q; q)ntn

=

∞k=0

q(d+1)

k2

(qd+1; qd+1)k

−q

(d−1)(d+2)2 td+1k ∞

n=0

(xt)n

(q; q)n

=

q

(d−1)(d+2)2 td+1

; qd+1∞

(xt; q)∞.

Applying the same technique with (2.20) to obtain (3.27). �

Remark 3.13. For d = 1, the identities (3.25) and (3.27) are reduced, respectively, to the knowngenerating relations [27, pp. 118–119]:

(t2; q2)∞

(xt; q)∞=

n≥0

hn(x |q)

(q; q)ntn,

(−xt; q)∞

(−t2; q2)∞=

n≥0

q( n2 )

(q; q)nhn(x |q)tn,

involving the discrete q-Hermite I and II polynomials.

3.6. Inversion formulas

Proposition 3.14. The inversion formula associated to the discrete q-Hermite I type polynomialsis given by:

xn=

[n

d+1 ]k=0

qk2 (d−1)(d+2)

(qd+1; qd+1)k

(q; q)n

(q; q)n−(d+1)kP(1)

n−(d+1)k(x |q). (3.28)

The discrete q-Hermite II type polynomials verify the following inversion formula:

xn=

[n

d+1 ]k=0

q(d+1)

k2

+

n−(d+1)k

2

−( n

2 )(−1)(d+1)k

(qd+1; qd+1)k

(q; q)n

(q; q)n−(d+1)kP(2)

n−(d+1)k(x |q). (3.29)

Proof. From (3.25) and (3.27), we have:

1(xt; q)∞

=1

q(d−1)(d+2)

2 td+1; qd+1∞

∞n=0

P(1)n (x |q)

(q; q)ntn,

(−xt; q)∞ = (−(−t)d+1; qd+1)∞

∞n=0

q( n2 )

(q; q)nP(2)

n (x |q)tn .


It follows from the identities (3.26) that

∞n=0

xn

(q; q)n

tn

=

∞n=0

[n

d+1 ]k=0

qk2 (d−1)(d+2)

(qd+1; qd+1)k(q; q)n−(d+1)kP(1)

n−(d+1)k(x |q)

tn,

∞n=0

q( n

2 )

(q; q)nxn

tn

=

∞n=0

[n

d+1 ]k=0

q(d+1)

k2

+

n−(d+1)k

2

(−1)(d+1)k

(qd+1; qd+1)k(q; q)n−(d+1)kP(2)

n−(d+1)k(x |q)

tn .

Then, by identification we get the desired result. �

3.7. d-dimensional functional vector

We recall that [22], a PS {Pn}n≥0 is d-orthogonal with respect to a d-dimensional functionalvector Γ =

t (Γ0,Γ1, . . . ,Γd−1) if and only if it is also d-orthogonal with respect to the vectorU =

t (u0, u1, . . . , ud−1), where the functionals u0, u1, . . . , ud−1 are the d first elements ofthe dual sequence {un}n≥0 associated to the PS {Pn}n≥0 and defined by ⟨ur , Pn⟩ = δn,r ; r ≥

0, n ≥ 0. Consequently, for the considered polynomials in this subsection, we determine the dfirst elements of the corresponding dual sequence to derive the d-dimensional functional vectorensuring the d-orthogonality of these polynomials. That leads to the following.

Proposition 3.15. The d-dimensional functional vectors ensuring the d-orthogonality of thediscrete q-Hermite I type polynomials and the discrete q-Hermite II type polynomials are,respectively, given by U (1)

=t (u(1)

0 , u(1)1 , . . . , u(1)

d−1) and U (2)=

t (u(2)0 , u(2)

1 , . . . , u(2)d−1) such that

⟨u(1)r , xn

⟩ = 0 and ⟨u(2)r , xn

⟩ = 0 if n < r, (3.30)

and if n ≥ r

⟨u(1)r , xn

⟩ = δr,i qk2 (d−1)(d+2) (q

r+1; q)(d+1)k

(qd+1; qd+1)k, (3.31)

⟨u(2)r , xn

⟩ = δr,i q(d+1)

k2

+( r

2 )−

r+(d+1)k2

(−1)(d+1)k (q; q)r+(d+1)k

(qd+1; qd+1)k(q; q)r, (3.32)

where n = i + (d + 1)k, k ∈ N, i = 0, 1, . . . , d, r = 0, 1, . . . , d − 1.

Proof. The identities (3.30) follow from the definition of a dual sequence.For n ≥ r , according to the inversion formulas given by (3.28) and (3.29), we have

⟨u(1)r , xn

⟩ =

[n

d+1 ]k=0

qk2 (d−1)(d+2)

(qd+1; qd+1)k

(q; q)n

(q; q)n−(d+1)k⟨u(1)

r , P(1)n−(d+1)k(x |q)⟩,

⟨u(2)r , xn

⟩ =

[n

d+1 ]k=0

q(d+1)

k2

+

n−(d+1)k

2

−( n

2 )(−1)(d+1)k

(qd+1; qd+1)k

×(q; q)n

(q; q)n−(d+1)k⟨u(2)

r , P(2)n−(d+1)k(x |q)⟩.

That, by virtue of the definition of a dual sequence, leads to (3.31) and (3.32). �


Remark 3.16. In the case when d = 1, the identity (3.31) is reduced to the following:⟨u(1)

0 , xn⟩ = 0, n = 2k + 1, k ∈ N,

⟨u(1)0 , xn

⟩ = (q; q2)k, n = 2k, k ∈ N.

Put n = 2k. Using the identities (3.26) and [27, p. 8]

(a; q)n =(a; q)∞

(aqn; q)∞, (3.33)

we obtain

⟨u(1)0 , xn

⟩ =(q; q2)∞

(q2k+1; q2)∞

= (q; q2)∞

∞j=0

(qn+1) j

(q2; q2) j.

That, by virtue of the identities (3.33) and [27, p. 9]:

(q2; q2) j = (q; q) j (−q; q) j

leads to

⟨u(1)0 , xn

⟩ =(q; q2)∞

(q, −q; q)∞

∞j=0

(q j+1, −q j+1; q)∞(qn+1) j . (3.34)

On the other hand, recall that the q-integral is defined by Koekoek and Swarttouw [27, p. 21]: z

0f (t)dq(t) := z(1 − q)

∞j=0

f (zq j )q j , z ∈ C.

Therefore 1

−1f (t)dq(t) = (1 − q)

∞j=0

[ f (q j ) − f (−q j )]q j .

Hence 1

−1(qt, −qt; q)∞xndq(t) = (1 − q)

∞j=0

(q j+1, −q j+1; q)∞[1 + (−1)n

](qn+1) j .

So that, for n = 2k + 1, we get 1

−1(qt, −qt; q)∞xndq(t) = 0,

and, for n = 2k, we obtain 1

−1(qt, −qt; q)∞xndq(t) = 2(1 − q)

∞j=0

(q j+1, −q j+1; q)∞(qn+1) j .

Using the identity (3.34), we deduce that

⟨u(1)0 , xn

⟩ =(q; q2)∞

2(1 − q)(q, −q; q)∞

1

−1(qt, −qt; q)∞xndq(t).


Thus, we rediscover the known functional ensuring the orthogonality of the discrete q-Hermitepolynomials given by Koekoek and Swarttouw [27, p. 118] 1

−1(qx, −qx; q)∞hm(x | q)hn(x | q)dq x = δm,n(1 − q)q( n

2 )(q; q)n(q, −1, −q; q)∞.

4. Concluding remarks

• The Hermite polynomials appeared in various characterization theorems for orthogonalpolynomials (see, for instance, [1,2,4,17,18,33,35]). In particular, Al-Salam [3] showed thatthe only OPSs of type (1.1) with d = 1 are the Hermite polynomials. In this paper, we give aq-extension of this result: Corollary 2.6. That may be considered, as far as we know, a furthercharacterization theorem of orthogonal polynomials not cited in the survey of Al-Salam [2]and Ismail’s book [26].

• In this work we introduce two new examples of d-orthogonal polynomials (see identities(2.19) and (2.20)), which may be viewed at the same time a generalization of the discrete q-Hermite polynomials I and II, and a q-extension of the d-orthogonal polynomials of Hermitetype known as the Gould–Hopper polynomials. That may be summarized by the followingscheme.

Acknowledgments

Sincere thanks are due to the referees for their careful reading of the manuscript and fortheir valuable comments and suggestions. The research was supported by the Ministry of HigherEducation and Technology, Tunisia (02/UR/1501).

References

[1] W.A. Al-Salam, On characterization of certain set of orthogonal polynomials, Boll. Unione Mat. Ital. 3 (1964)448–450.

[2] W.A. Al-Salam, Characterisation Theorems for Orthogonal Polynomials, Columbus, OH, 1989, in: NATO Adv. Sci.Inst. Ser. C Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 1–24.

[3] N. Al-Salam, Orthogonal polynomials of hypergeometric type, Ser. Math. Inform. 33 (1966) 109–121.[4] A. Angelesco, Sur les polynomes orthogonaux en rapport avec d’autres polynomes, Buletinul Societatii Stiite din

Cluj 1 (1921) 44–59.[5] Y. Ben Cheikh, N. Ben Romdhane, d-Orthogonal polynomial sets of Tchebytchev type, in: S. Elaydi, et al.

(Eds.), Proceedings of the International Conference on Difference Equations, Special Functions and OrthogonalPolynomials, Munich, Germany 25–30 July, World Scientific, 2005, pp. 100–111.


[6] Y. Ben Cheikh, K. Douak, On the classical d-orthogonal polynomials defined by certain generating functions I,Bull. Belg. Math. Soc. 7 (2000) 107–124.

[7] Y. Ben Cheikh, K. Douak, On the classical d-orthogonal polynomials defined by certain generating functions II,Bull. Belg. Math. Soc. 8 (2001) 591–605.

[8] Y. Ben Cheikh, K. Douak, A generalized hypergeometric d-orthogonal polynomial set, C. R. Acad. Sci. Paris 331(2000) 349–354.

[9] Y. Ben Cheikh, K. Douak, On two-orthogonal polynomials related to Bateman’s J u,vn -function, Methods Appl.

Anal. 7 (2000) 641–662.[10] Y. Ben Cheikh, M. Gaied, Dunkl–Appell d-orthogonal polynomials, Integral Transforms Spec. Funct. 18 (2007)

581–597.[11] Y. Ben Cheikh, I. Lamiri, A. Ouni, On Askey-scheme and d-orthogonality, I: a characterization theorem, J. Comput.

Appl. Math. 233 (2009) 621–629.[12] Y. Ben Cheikh, I. Lamiri, A. Ouni, d-orthogonality of little q-Laguerre type polynomials, J. Comput. Appl. Math.

236 (2011) 7484.[13] Y. Ben Cheikh, A. Ouni, Some generalized hypergeometric d-orthogonal polynomial sets, J. Math. Anal. Appl. 343

(2008) 464–478.[14] Y. Ben Cheikh, A. Zaghouani, Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math. 156 (2003)

253–263.[15] Y. Ben Cheikh, A. Zaghouani, d-orthogonality via generating functions, J. Comput. Appl. Math. 199 (2007) 2–22.[16] N. Ben Romdhane, d-orthogonal Faber polynomials, Integral Transforms Spec. Funct. 18 (2007) 663–677.[17] L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Port. Math. 20 (1961) 43–46.[18] T.S. Chihara, Orthogonal polynomials with Brenke type generating function, Duke Math. J. 35 (1968) 505–518.[19] K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70

(1996) 279–295.[20] K. Douak, P. Maroni, On d-orthogonal Tchebyshev polynomials I, Appl. Numer. Math. 24 (1997) 23–53.[21] K. Douak, P. Maroni, On d-orthogonal Tchebyshev polynomials II, Methods Appl. Anal. 4 (1997) 404–429.[22] K. Douak, P. Maroni, Une caracterisation des polynomes classiques de dimension d , J. Approx. Theory 82 (1995)

177–204.[23] K. Douak, P. Maroni, Les polynomes orthogonaux “classiques” de dimension deux, Analysis 12 (1992) 71–107.[24] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke

Math. J. 29 (1962) 51–63.[25] W. Hahn, Uber orthogonalpolynome die q-differenzengleichungen genugen, Math. Nachr. 2 (1949) 4–34.[26] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Canbridge University Press, New

York, 2005.[27] R. Koekoek, R.F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue,

Technical Report 98-17, Faculty of the Technical Mathematics and Informatics, Delft University of Technology.,Delft, 1998.

[28] I. Lamiri, A. Ouni, d-orthogonality of Humbert and Jacobi type polynomials, J. Math. Anal. Appl. 341 (2008)24–51.

[29] I. Lamiri, A. Ouni, d-orthogonality of Hermite type polynomials, App. Math. Comput. 202 (2008) 24–43.[30] Y.L. Luke, The Special Functions and their Approximations, Vol. I, Academic Press, New York, San Francisco,

London, 1969.[31] P. Maroni, l’orthogonalite et les recurrences de polynomes d’ordre superieur a deux, Ann. Fac. Sci. Toulouse Math.

10 (1989) 105–139.[32] P. Maroni, Two-dimensional orthogonal polynomials, their associated sets and the co-recursive sets, Number.

Algorithms 3 (1992) 299–312.[33] J. Meixner, Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion, J. Lond. Math.

Soc. 9 (1934) 6–13.[34] J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math. 19 (1987) 141–150.[35] K.P. Williams, A uniqueness theorem for the Legendre and Hermite polynomials, Trans. Amer. Math. Soc. 26

(1924) 441–445.[36] A. Zaghouani, Some basic d-orthogonal polynomial sets, Georgian Math. J. 12 (2005) 583–593.

Download - -orthogonality of discrete -Hermite type polynomials

Top Related