ON SOME PROPERTIES OF A CLASS OFPOLYNOMIALS SUGGESTED BY MITTAL

A. K. SHUKLAand

J. C. PRAJAPATI

S. V. NATIONAL INSTITUTE OF TECHNOLOGY, INDIA

Received : September 2006. Accepted : March 2007

ProyeccionesVol. 26, No 2, pp. 145-156, August 2007.Universidad Catolica del NorteAntofagasta - Chile

Abstract

The object of this paper is to establish some generating relations

by using operational formulae for a class of polynomials T(α+s−1)kn (x)

defined by Mittal. We have also derived finite summation formulae for(1.6) by employing operational techniques. In the end several specialcases are discussed.

Key Words : Operational formulae; generating relations; finitesum formulae.

2000 Mathematics Subject Classification : 33E12; 33E99;44A45.

146 A. K. Shukla and J. C. Prajapati

1. Introduction

Chak [1] defined a class of polynomials as:

G(α)n,k(x) = x−α−kn+nex(xkD)n[xαe−x](1.1)

where D =d

dx, k is constant and n = 0, 1, 2, . . . .

Chatterjea [2] studied a class of polynomials for generalized Laguerrepolynomial as:

T (α)rn (x, p) =1

n!x−α−n−1 exp(pxr)(x2D)n[xα+1 exp(−pxr)].(1.2)

Gould and Hopper [3] introduced generalized Hermite polynomials as:

Hrn(x, a, p) = (−1)nx−a exp(pxr)Dn[xa exp(−pxr)].(1.3)

Singh [10] obtained generalized Truesdell polynomials by using Ro-drigues formula, which is defined as:

T (α)n (x, r, p) = x−α exp(pxr)(xD)n[xα exp(−pxr)].(1.4)

In 1971, Mittal [5] proved the Rodrigues formula for a class of polyno-

mials T(α)kn (x) as:

T(α)kn (x) =

1

n!x−α exp{pk(x)}Dn[xα+n exp{−pk(x)}](1.5)

where pk(x) is a polynomial in x of degree k.

Mittal [6] also proved the following relation for (1.5)

T(α+s−1)kn (x) =

1

n!x−α−n exp{pk(x)}θn[xα exp{−pk(x)}](1.6)

and an operator θ ≡ x(s+ xD), where s is constant.

The following well-known facts are prepared for studying (1.6).

Generalised Laguerre polynomials (Srivastava and Manocha[12]) defined as:

L(α)n (x) =x−α−n−1 ex

n!(x2D)n[xα+1 e−x].(1.7)

On Some Properties of a Class of Polynomials Suggested by Mittal 147

Hermite polynomials (Rainville [9]) defined as:

Hn(x) = (−1)n exp(x2) Dn[exp(−x2)].(1.8)

Konhauser polynomials of first kind (Srivastava [11]) defined as:

Y αn (x; k) =

x−kn−α−1 ex

kn n!(xk+1D)n[xα+1 e−x].(1.9)

Konhauser polynomials of second kind (Srivastava [11]) defined as:

Zαn (x; k) =

Γ(kn+ α+ 1)

n!

nXj=0

(−1)jÃ

nj

!xkj

Γ(kj + α+ 1)(1.10)

where k is a positive integer.

Srivastava and Manocha [12] verified following result by using induc-tion method,

(x2D)n{f(x)} = xn+1Dn{xn−1f(x)}.(1.11)

2. Definitions and Notations

McBride [4] defined generating function as:

Let G(x, t) be a function that can be expanded in powers of t suchthat

G(x, t) =∞Pn=0

cnfn(x)tn, where cn is a function of n that may contain

the parameters of the set {fn(x)}, but is independent of x and t. ThenG(x, t) is called a generating function of the set {fn(x)}.

Remark: A set of functions may have more than one generating function.

In our investigation we used the following properties of the differentialoperators;

θ ≡ x(s + xD) and θ1 ≡ (1 + xD), where D ≡ d

dx, (Mittal [7], Patil

and Thakare [8]) which are useful to establish linear generating relationsand finite sum formulae.

148 A. K. Shukla and J. C. Prajapati

(2.1) θn = xn(s+ xD)(s+ 1 + xD)(s+ 2 + xD) . . . (s+ (n− 1) + xD)

(2.2) θn(xα) = (α+ s)n xα+n

(2.3) θn(xuv) = x∞X

m=0

Ãnm

!θn−m(v)θm1 (u)

(2.4) etθ(xα) = xα(1− xt)−(α+s)

(2.5) etθ(xuv) = xetθ(v)etθ1(u)

(2.6) etθ(xαf(x)) = xα(1− xt)−(α+s) f∙x(1− xt)−1

¸

(2.7) etθ(xα−nf(x)) = xα(1 + t)−1+(α+s) f∙x(1 + t)

¸

(2.8) (1− at)−α/a = (1− at)−β/a∞X

m=0

µα− β

a

¶m

(at)m

m!

3. Generating Relations

We obtained some generating relations of (1.6) as

∞Xn=0

T(α+s−1)kn (x)tn = (1− t)−(α+s) exp[pk(x)− pk{x(1− t)−1}](3.1)

On Some Properties of a Class of Polynomials Suggested by Mittal 149

∞Xn=0

T(α−n+s−1)kn (x)tn = (1 + t)−1+(α+s) exp[pk(x)− pk{x(1 + t)}]

(3.2)

∞Pm=0

Ãm+ nn

!T(α+s−1)k(n+m) (x) t

m

= (1− t)−(α+s+n) exphpk (x)− pk

nx (1− t)−1

oiT(α+s−1)kn

nx (1− t)−1

o(3.3)

∞Pm=0

Ãm+ nn

!T(α−m+s−1)k(n+m) (x) tm

= (1 + t)α+s−1 exp [pk (x)− pk {x (1 + t)}]T (α−m+s−1)kn {x (1 + t)}

(3.4)

Proof of (3.1). From (1.6), we consider

∞Xn=0

xn T(α+s−1)kn (x)tn = x−α exp{pk(x)} etθ [xα exp{−pk(x)}]

and using (2.6), above equation reduces to,

∞Xn=0

xn T(α−s+1)kn (x)tn = x−α exp{pk(x)}xα (1−xt)−(α+s) exp[−pk{x(1−xt)−1}]

= (1− xt)−(α+s) exp[pk(x)− pk{x(1− xt)−1}]

replacing t by t/x, which gives (3.1).

150 A. K. Shukla and J. C. Prajapati

Proof of (3.2). From (1.6) we consider,

T(α−n+s−1)kn (x) =

1

n!x−(α−n)−n exp{pk(x)} θn [xα−n exp{−pk(x)}]

or

∞Xn=0

T(α−n+s−1)kn (x)tn = (x)−α exp{pk(x)} etθ [xα−n exp(−pk(x))]

by using (2.7), we get

∞Xn=0

T(α−n+s−1)kn (x)tn = x−α exp{pk(x)}xα(1+t)−1+(α+s) exp{−pk{x(1+t)}]

= (1 + t)−1+(α+s) exp[pk(x)− pk{x(1 + t)}].

Proof of (3.3). Again from (1.6) we consider,

θn[xα exp{−pk(x)}] = n! xα+n exp{−pk(x)}T (α+s−1)kn (x)

or

etθ(θn[xα exp{−pk(x)}]) = n! etθ [xα+n exp{−pk(x)}T (α+s−1)kn (x)]

using (2.6) we get,

∞Xm=0

tm θm+n

m![xα exp{−pk(x)}]

= n! xα+n(1− xt)−(α+s+n) exp[−pk{x(1− xt)−1}]T (α+s−1)kn {x(1− xt)−1}

therefore, we get

On Some Properties of a Class of Polynomials Suggested by Mittal 151

∞Xm=0

1

m! n!(m+n)! xα+m+n exp{−pk(x)}T (α+s−1)k(m+n) (x)t

m

= xα+n(1− xt)−(α+s+n) exp[−pk{x(1− xt)−1}]T (α+s−1)kn {x(1− xt)−1}

hence above equation reduces to,

∞Xm=0

xmÃ

m+ nn

!T(α+s−1)k(m+n) (x)t

m

= (1− xt)−(α+s+n) exp[pk(x)− pk{x(1− xt)−1}]T (α+s−1)kn {x(1− xt)−1}

replacing t by t/x, which gives (3.3).

Proof of (3.4). Again from (1.6) we consider,

θn[xα exp{−pk(x)}] = n! xα+n exp{−pk(x)}T (α+s−1)kn (x)

replacing α by α−m, we get

θn[xα−m exp{−pk(x)}] = n! xα−m+n exp{−pk(x)}T (α−m+s−1)kn (x)

or

etθ(θn[xα−mEα{−pk(x)}]) = n! etθ[x(α+n)−m exp{−pk(x)}T (α−m+s−1)kn (x)]

using (2.7) we get,

∞Xn=0

tm θm+n

m![xα−m exp{−pk(x)}]

= n! xα+n(1 + t)α+s−1 exp[−pk{x(1 + t)}]T (α−m+s−1)kn {x(1 + t)}

152 A. K. Shukla and J. C. Prajapati

therefore, we get

∞Xm=0

1

m! n!(m+n)! xα−m+m+n exp{−pk(x)}T (α−m+s−1)k(m+n) (x)tm

= xα+n(1 + t)α+s−1 exp[−pk{x(1 + t)}]T (α−m+s−1)kn {x(1 + t)}

which reduces to (3.4).

4. Finite Summation Formulae

We obtained finite summation formula for (1.6) as

T(α+s−1)kn (x) =

nXm=0

(m!)−1 (α− β)m T(β+s−1)k(n−m) (x)(4.1)

T(α+s−1)kn (x) =

nXm=0

1

m!(α)m T

(s−1)k(n−m)(x)(4.2)

Proof of (4.1). We can write (1.6) as,

∞Xn=0

xn T(α+s−1)kn (x)tn = x−α exp{pk(x)} etθ[xα exp{−pk(x)}]

by using (2.6), we write

∞Xn=0

xnT(α+s−1)kn (x) tn

= x−α exp{pk(x)}xα(1− xt)−(α+s) exp[−pk{x(1− xt)−1}]

= (1− xt)−(α+s) exp[pk(x)− pk{x(1− xt)−1}]

applying (2.8), which yields

∞Xn=0

xn T(α+s−1)kn (x)tn

On Some Properties of a Class of Polynomials Suggested by Mittal 153

= (1− xt)−(β+s)∞X

m=0

(α− β)m(xt)m

m!exp[pk(x)− pk{x(1− xt)−1}]

=∞Xn=0

(α− β)mxm tm

m!exp{pk(x)}(1− xt)−(β+s) exp[−pk{x(1− xt)−1}]

using (3.1), above equation reduces to,X∞n=0

xn T(α+s−1)kn (x)tn =

=∞X

m=0

(α− β)mxm tm

m!exp{pk(x)}x−β etθ[xβ exp{−pk(x)}]

=∞X

m,n=0

(α− β)mxm tn+m

m! n!exp{pk(x)}x−β θn[xβ exp(−pk(x))]

=∞Xn=0

nXm=0

1

m!(α− β)m

x−β+m

(n−m)!exp{pk(x)} θn−m[xβ exp{−pk(x)}]tn

equating the coefficients of tn, we get

xnT(α+s−1)kn (x) =

nXm=0

1

m!(α−β)m

x−β+m

(n−m)!exp{pk(x)} θn−m[xβ exp{−pk(x)}]

Therefore, we obtain

T(α+s−1)kn (x) =

nXm=0

1

m!(α−β)m

x−β(−n−m)

(n−m)!exp{pk(x)} θn−m[xβ exp{−pk(x)}]

and applying (1.6) then above equation immediately leads to (4.1).

Proof of (4.2). We can write (1.6) as,

T(α+s−1)kn (x) =

1

n!x−α−n exp{pk(x)} θn[xxα−1 exp{−pk(x)}]

154 A. K. Shukla and J. C. Prajapati

using (2.3) we get,

and by using (2.1) which yields,

T(α+s−1)kn (x) =

1

n!x−α−n exp{pk(x)}x

nXm=0

n!

m! (n−m)!

×xn−m[(s+xD)(s+1+xD)(s+2+xD) . . . (s+(n−m−1)+xD)] exp{−pk(x)}

×xm[(1+xD)(2+xD)(3+xD) . . . (m+xD)]xα−1

T(α+s−1)kn (x) = exp{pk(x)}

nXm=0

1

m! (n−m)!

n−m−1Yi=0

(s+i+xD) exp{−pk(x)}(α)m

(4.3)

Putting α = 0 and replacing n by n−m in (1.6) which reduces to

T(s−1)k(n−m)(x) =

1

(n−m)!x−(n−m) exp{pk(x)} θn−m[exp{−pk(x)}]

thus, we have

1

(n−m)!θn−m[exp{−pk(x)}] =

xn−m

exp{pk(x)}T(s−1)k(n−m)(x)

using (2.1), we get

1

(n−m)!

n−m−1Yi=0

(s+ i+ xD)[exp{−pk(x)}] =1

exp{pk(x)}T(s−1)k(n−m)(x).

(4.4)

use of (4.4) and (4.3), gives complete proof of (4.2).

On Some Properties of a Class of Polynomials Suggested by Mittal 155

5. Concluding Remarks

Some special cases of T(α+s−1)kn (x) polynomials are given below:

If we replace α by α + 1, pk(x) = p1(x) = x and s = 0 in (1.6), thenthis equation reduces to

T (α)n (x) = L(α)n (x) = Zαn (x; 1) = Y α

n (x; 1).(5.1)

Again replacing α by α + 1, pk(x) = pxr and s = 0 in (1.6), whichgives

T (α)rn (x) = T (α)rn (x, p).(5.2)

Substituting α = 1 − n, pk(x) = x2, s = 0 in (1.6) and using (1.11),which yields

T(1−n)2n (x) =

(−x)nn!

Hn(x).(5.3)

AcknowledgementsWe express our sincere thanks to the referees for their kind comments

for the improvement of this manuscript.

References

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156 A. K. Shukla and J. C. Prajapati

[6] H. B. Mittal, Operational representations for the generalized Laggurepolynomial, Glasnik Mat. Ser III, 6(26), pp. 45—53, (1971).

[7] H. B. Mittal, Bilinear and bilateral generating relations, American J.Math. 99, pp. 23—45, (1977).

[8] K. R. Patil and N.K. Thakre, Operational formulas for a functiondefined by a generalized Rodrigues formula-II, Sci. J. Shivaji Univ.15, pp. 1—10, (1975).

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[10] R. P. Singh, On generalized Truesdell polynomials, Rivista de Math-ematica, 8, pp. 345—353, (1968).

[11] H. M. Srivastava, Some biorthogonal polynomials suggested by theLaguerre polynomials, Pacific J. Math. 98(1), pp. 235—250, (1982).

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A. K. ShuklaDepartment of MathematicsS. V. National Institute of TechnologySurat-395007Indiae-mail : ajayshukla2@rediffmail.com

and

J. C. PrajapatiDepartment of MathematicsS. V. National Institute of TechnologySurat-395007Indiae-mail : jyotindra18@rediffmail.com