South Asian Journal of Mathematics
http://www.sajm.com.nu
2013 , Vol. 3 ( 3 ) : 199 ∼ 210
ISSN 2251-1512
RESEARCH ARTICLE
On new sequence of functions involving pFq Praveen Agarwal
¬∗
, Mehar Chand
¬ Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India Department of Mathematics, Malwa College of IT and Management, Bathinda-151001, India E-mail:
[email protected] Received: Mar-24-2013; Accepted: Apr-29-2013 *Corresponding author
Abstract A remarkably large number of operational techniques have drawn the attention of several researchers in the study of sequence of functions and polynomials. In this sequel, here, we aim to introduce a new sequence of functions involving the p Fq by using operational techniques. Some generating relations and finite summation formulae of the sequence presented here are also considered. In the last, we use Matlab(R2012a) for each parameters of our main sequence, which gives the eccentric charecteristics in the area of sequence of functions or class of polynomials. Key Words MSC 2010
1
special function, generating relations, gaussian hypergeomtric functions, sequence of function 33E10, 44A45
Introduction
The idea of representing the processes of calculus, derivation, and integration, as operators is called operational technique, it is also known as operational Calculus . Many operational technique involving various special functions, have found significant importance and applications in various subfield of applicable mathematical analysis. Several applications of operational techniques can be found in the problems in analysis, in particular differential equations are transformed into algebraic problems, usually the problem of solving a polynomial equations.Since last four decades, a number of workers like Chak[1],Gould and Hopper [5], Chatterjea[4], Singh[15],Srivastava and Singh[18], Mittal[7,8,9], Chandal[2,3], Srivastava[13], Joshi and Parjapat[6],Patil and Thakare[10] and Srivastava and Singh[17] have studied in depth, the properties, applications and different extensions of the various operational technique. d The key element of the operational technique is to consider differentiation as an operator D = dx acting on functions. Linear differential equations can then be recast in the form of an operator valued function F (D) of the operator D acting on the unknown function equals the known function. Solutions are then obtained by making the inverse operator of F act on the known function. Indeed, a remarkably large number of sequence of functions involving a variety of special functions have been developed by many authors (see, for example, [17]; for a very recent work, see also [15]).Here we aim at presenting a new sequence of functions involving the p Fq by using operational techniques,which are expressed in terms of the gauss hypergeometric function.Some generating relations and finite summation formulae are also obtained.
Citation: Praveen Agarwal, Mehar Chand, On new sequence of functions involving 199-210.
p Fq ,
South Asian J Math, 2013, 3(3),
P. Agarwal, et al : On new sequence of functions involving p Fq
For our purpose, we begin by recalling some known functions and earlier works. In 1971,Mittal [7] gives the Rodrigues formula for the generalized Lagurre polynomials defined as: (α)
Tkn (x) =
£ ¤ 1 −α x exp (pk (x)) Dn xα+n exp (−pk (x)) n!
(1.1)
where pk (x) is a polynomial in x of degree k. Mittal [8] also proved the following relation for (1.1) defined as: (α+s−1)
Tkn
(x) =
1 −α−n x exp (pk (x)) Tsn [xα exp (−pk (x))] n!
(1.2)
where s is constant and Ts ≡ x (s + xD). (α)
In this sequel, in 1979, Srivastava and Singh [17] studied a sequence of functions Vn as: Vn(α) (x; a, k, s) =
(x; a, k, s) defined
x−α exp {pk (x)} θn [xα exp {−pk (x)}] n!
(1.3)
By employing the operator θ ≡ xa (s + xD), where s is constant and pk (x)is a polynomial in x of degree k. o n (p;q;α) (x; a, k, s) /n = 0, 1, 2, ... is introduced in this paper as: A new sequence of function Vn Vn(p,q,α) (x; a, k, s) (a ) ; 1 p n − pk (x) xα p Fq = x−α p Fq pk (x) (Txa,s ) n! (bq ) ; (bq ) ;
(ap ) ;
(1.4)
d where Txa,s ≡ xa (s + xD) , D ≡ , a and s are constants, β > 0, k is finite and non-negative integer, dx pk (x) is a polynomial in x of degree k and p Fq is a special cases of the generalized hypergeometric functions of one variables, for the sake of completeness we define this function here. A generalized hypergeometric function p Fq is defined and represented as follows(see [14,Section 1.5]): p Fq
(ap ) ; (bq ) ;
z =
∞ X n=0
Qp Qj=1 q j=1
(aj )n z n (bj )n n!
, provided p 6 q; p = q + 1and |z| < 1.
(1.5)
where (λ)n is the Pochhammer symbol defined (for λ ∈ C) by (see [14,p. 2 and pp. 4-6]): ( (λ)n : =
1
(n = 0)
λ(λ + 1) . . . (λ + n − 1) (n ∈ N := {1, 2, 3, . . .})
Γ(λ + n) = Γ(λ)
(λ ∈ C \
(1.6)
Z− 0)
and Z− 0 denotes the set of nonpositive integers. Some generating relations and finite summation formulae of class of polynomials or sequence of function have been obtained by using the properties of the differential 200
South Asian J. Math. Vol. 3 No. 3
d operators. Txa,s ≡ xa (s + xD) , Txa,1 ≡ xa (1 + xD), where D ≡ , is based on the work of Mittal [9], dx Patil and Thakare [10], Srivastava and Singh [17]. Some useful Operational Techniques are given below: ³ ´ ¡ ¢ −1/a − β+s exp (tTxa,s ) xβ f (x) = xβ (1 − axa t) ( a ) f x (1 − axa t) (1.7) ´ ³ ¡ ¢ −1+( α+s 1/a a ) exp (tTxa,s ) xα−an f (x) = xα (1 + at) f x (1 + at)
(1.8)
¢ ¡ n n (Txa,s )n−m (v) Txa,1 m (u) (Txa,s ) (xuv) = x m=0 m
(1.9)
∞ X
(1 + xD) (1 + a + xD) (1 + 2a + xD) (1 + 3a + xD) ... (1 + (m − 1)a + xD) xβ−1
= am
(1 − at)
2
−α a
µ ¶ β xβ−1 a m
= (1 − at)
−β a
(1.10)
¶ ∞ µ m X α−β (at) a m m! m=0
(1.11)
Generating Relations First generating relation: ∞ X
Vn(p,q,α) (x; a, k, s) x−an tn
n=0
− = (1 − at) (
α+s a
) F p q
(ap ) ; (bq ) ;
pk (x) p Fq
(ap ) ; (bq ) ;
³
−1/a
´
− pk x (1 − at)
(2.12)
Second generating relation: ∞ X
Vn(p,q,α−an) (x; a, k, s) x−an tn
n=0
−1+( α+s a )
= (1 + at)
p Fq
(ap ) ; (bq ) ;
; pk (x) p Fq
(ap ) ; (bq ) ;
³
1/a
− pk x (1 + at)
´
(2.13)
Third generating relation: ∞ X m+n Vn(p,q,α) (x; a, k, s) x−an tn m n=0 201
P. Agarwal, et al : On new sequence of functions involving p Fq
p Fq − = (1 − at) (
α+s a
)
(bq ) ;
p Fq
(ap ) ;
(ap ) ;
(bq ) ;
pk (x)
³
−1/a
pk x (1 − at)
´
³ ´ Vn(p,q,α) x (1 − at)−1/a ; a, k, s
(2.14)
Proof of first generating relation: We start from (1.4) and consider: ∞ X
Vn(p,q,α) (x; a, k, s) tn
n=0
(a ) ; p = x−α p Fq pk (x) exp(tTxa,s ) xα p Fq − pk (x) (bq ) ; (bq ) ;
(ap ) ;
(2.15)
Using operational technique (1.7), above equation (2.4) reduces to: ∞ X
Vn(p,q,α) (x; a, k, s) tn
n=0
= x−α p Fq
(ap ) ;
pk (x) xα (1 − axa t)−(
(bq ) ;
= (1 − axa t)−(
α+s a
) F p q
α+s a
) F p q
(ap ) ; (bq ) ;
(ap ) ; (bq ) ;
pk (x) p Fq
(ap ) ; (bq ) ;
³
´
− pk x(1 − axa t)−1/a
³
´
− pk x(1 − axa t)−1/a
(2.16)
and replacing t by tx−a , this gives (2.1). Proof of second generating relation: Again from (1.4), we have: ∞ X
x−an Vn(p,q,α−an) (x; a, k, s) tn
n=0
(a ) ; p = x−α p Fq pk (x) exp(tTxa,s ) xα−an p Fq − pk (x) (bq ) ; (bq ) ;
(ap ) ;
applying the operational technique (1.8), we get: ∞ X n=0
202
x−an Vn(p,q,α−an) (x; a, k, s) tn
(2.17)
South Asian J. Math. Vol. 3 No. 3
= x−α p Fq
(ap ) ; (bq ) ;
pk (x) xα (1 + at)
= (1 + at)
α+s a −1
p Fq
α+s a −1
(ap ) ; (bq ) ;
p Fq
(ap ) ; (bq ) ;
pk (x) p Fq
(ap ) ; (bq ) ;
³
1/a
− pk x (1 + at)
³
1/a
− pk x (1 + at)
´
´
(2.18)
(2.19)
This proves (2.2). Proof of third generating relation: We can write (1.4) as:
n (Txa,s ) xα p Fq
(ap ) ; (bq ) ;
− pk (x)= n!xα
p Fq
1 (ap ) ; (bq ) ;
Vn(p,q,α) (x; a, k, s)
(2.20)
pk (x)
Or (a ) ; p n − pk (x) exp (t (Txa,s )) (Txa,s ) xα p Fq (bq ) ;
1 α (p,q,α) x = n! exp (tTxa,α ) V (x; a, k, s) n (a ) ; p pk (x) p Fq (bq ) ;
(2.21)
∞ m X (ap ) ; t m+n xα p Fq − pk (x) (Txa,s ) m! (bq ) ; m=0 1 a,s (p,q,α) α Vn = n! exp (tTx ) x (x; a, k, s) (ap ) ; F p (x) p q k (bq ) ;
(2.22)
Using the operational technique (1.7), above equation can be written as: ∞ X (ap ) ; tm a,s m+n α (T ) x p Fq − pk (x) m! x (bq ) ; m=0 203
P. Agarwal, et al : On new sequence of functions involving p Fq
− α+s = n!xα (1 − axa t) ( a ) p Fq
³ ´ × Vn(p,q,α) x (1 − axa t)−1/a ; a, k, s (2.23)
1
³ ´ −1/a pk x (1 − axa t)
(ap ) ; (bq ) ;
use of (2.9) gives: ∞ X tm (m + n)! α x m!n! m=0
− α+s = xα (1 − axa t) ( a )
p Fq
(ap ) ;
(bq ) ;
(p,q,α) Vm+n (x; a, k, s)
− pk (x)
³ ´ Vn(p,q,α) x (1 − axa t)−1/a ; a, k, s (2.24)
1
p Fq
1
³ ´ −1/a − pk x (1 − axa t)
(ap ) ; (bq ) ;
Therefore ∞ X
m+n n
m=0
(p,q,α) Vm+n (x; a, k, s) tm
p Fq − = (1 − axa t) (
α+s a
)
p Fq
(ap ) ; (bq ) ;
(ap ) ; (bq ) ;
pk (x)
³
−1/a
pk x (1 − axa t)
´
³ ´ Vn(p,q,α) x (1 − axa t)−1/a ; a, k, s
(2.25)
And replacing t by tx−a , this gives the result (2.3). Remark 1. If we give some suitable parametric replacement in (2.1), (2.2) and (2.3) respectively, then we can arrive at the known results (see [1,2,3,4,5,6,7,8,10,12,13,15,16]).
3
Finite Summation Formulas
First finite summation formulae: ∞ ³ ´ X 1 (p,q,0) m α Vn−m (x; a, k, s) (axa ) m! a m m=0
(3.26)
µ ¶ ∞ X 1 α−β (p,q,β) m (axa ) Vn−m (x; a, k, s) m! a m m=0
(3.27)
Vn(p,q,α) (x; a, k, s) = Second finite summation formulae: Vn(p,q,α) (x; a, k, s) =
Proof of first finite summation formulae:
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South Asian J. Math. Vol. 3 No. 3
From equation (1.4), we have: (a ) ; 1 p n Vn(p,q,α) (x; a, k, s)= x−α p Fq pk (x) (Txa,s ) xxα−1 p Fq − pk (x) n! (bq ) ; (bq ) ;
(ap ) ;
(3.28)
Using the operational technique (1.9), we have: ∞ X (a ) ; n 1 p (Txa,s )n−m Vn(p,q,α) (x; a, k, s) = x−α p Fq pk (x) x n! (bq ) ; m=0 m
¡ ¢m ¡ α−1 ¢ − pk (x) Txa,1 x p Fq (bq ) ;
(ap ) ;
∞ X (ap ) ; n! 1 −α pk (x) x xa(n−m) × = x p Fq n! m! (n − m)! (bq ) ; m=0 [(s + xD) (s + a + xD) (s + 2a + xD) ... (s + (n − m − 1) a + xD)] p Fq
(ap ) ; (bq ) ;
− pk (x) xam ×
¡ ¢ [(1 + xD) (1 + a + xD) (1 + 2a + xD) ... (1 + (m − 1) a + xD)] xα−1
(3.29)
Using the result (1.9), we have: n n−m−1 ³α´ X Y (ap ) ; (a ) ; 1 p pk (x) − pk (x) am = p Fq (s + ia + xD) p Fq xan × m! (n − m)! a m (bq ) ; (bq ) ; m=0 i=0 Put α = 0 and replacing n by n − m in (3.3), we get:
(p,q,0) Vn−m
(ap ) ; (a ) ; 1 p n−m (x; a, k, s)= pk (x) (Txa,s ) − pk (x) p Fq p Fq (n − m)! (bq ) ; (bq ) ;
(a ) ; 1 p n−m ⇒ (Txa,s ) − pk (x) = p Fq (n − m)! (bq ) ;
This gives 1 (n − m)!
n−m−1 Y i=0
p Fq
1 (ap ) ; (bq ) ;
(s + ia + xD) p Fq − pk (x) = xa(m−n) (bq ) ;
(3.30)
(p,q,0) Vn−m (x; a, k, s)
pk (x)
(ap ) ;
p Fq
1 (ap ) ; (bq ) ;
(p,q,0) Vn−m (x; a, k, s)
pk (x)
205
P. Agarwal, et al : On new sequence of functions involving p Fq
From equation (3.5) and (3.8) we have the main the result. Proof of second finite summation formulae: Equation (1.4) can be written as: ∞ ´ ³ X (a ) ; (a ) ; p p Vn(p,q,α) (x; a, k, s) tn = x−α p Fq pk (x) exp tTx(a,s) xα p Fq − pk (x) (3.31) (bq ) ; (bq ) ; n=0
Applying the (1.7) in equation (3.9), we have: ∞ X
Vn(p,q,α) (x; a, k, s) tn
n=0
= x−α p Fq
(ap ) ; (bq ) ;
−( α+s a )
pk (x) xα (1 − axa t)
− = (1 − axa t) (
α+s a
) F p q
(ap ) ; (bq ) ;
p Fq
(ap ) ; (bq ) ;
pk (x) p Fq
(ap ) ; (bq ) ;
³
−1/a
− pk x (1 − axa t)
³
−1/a
− pk x (1 − axa t)
´
´
(3.32)
Applying the result from equation (1.11); equation (3.10) reduces to:
−( β+s a )
= (1 − axa t)
¶ ∞ µ m ³ ´ X (ap ) ; (ap ) ; α−β (axa t) −1/a pk (x) p Fq − pk x (1 − axa t) p Fq a m! (b ) ; (b ) ; m q q m=0
¶ ∞ µ a m X (a ) ; (a ) ; α−β (ax t) −β p p x p Fq pk (x) exp (tTxa,s ) xβ p Fq − pk (x) = a m! (bq ) ; (bq ) ; m m=0
=
¶ ∞ µ ∞ X X α−β m=0 n=0
a
m
m (a ) ; (ap ) ; (axa ) tn+m −β p n − pk (x) xβ p Fq pk (x) (Txa,s ) x p Fq m!n! (bq ) ; (bq ) ;
¶ ∞ X n µ a m n X (ap ) ; (ap ) ; α−β (ax ) t n−m = x−β p Fq pk (x) (Txa,s ) xβ p Fq − pk (x) a (bq ) ; (bq ) ; m m! (n − m)! n=0 m=0 Now equating the coefficient of tn , we get: Vn(p,q,α)
¶ n µ m X (ap ) ; (axa ) α−β n−m −β x p Fq (x; a, k, s) = pk (x) (Txa,s ) a m! (n − m)! (bq ) ; m m=0 (a ) ; p xβ p Fq − pk (x) (bq ) ;
Using the equation (1.4) in (3.13), we have the result (3.2).
206
(3.33)
South Asian J. Math. Vol. 3 No. 3
4
Special Cases
(I) If we apply the case of Mittage-Leffler function via hypergeometric function i.e. Eα = 0 Fα−1 ¢ ; ; all the results estabished in equation (2.1), (2.2), (2.3), (3.1) and (3.2) reduced to the work of Ibrahim A. Salehbhai, Jyotindra C. Prajapati and Ajay K. Shukla [12].
¡
1 2 α−1 z α , α , ..., α ; αα ,
(II) If we apply the wright function W (α, δ; z) very special case of hypergeomtric function p Fq ; all the results estabished in equation (2.1), (2.2), (2.3), (3.1) and (3.2) reduced to the work of J. C. Prajapati and N. K. Ajudia [11].
5
Matlab implementation
The matlab is one of the important aspect mainly in the field of science and engineering. Therefore the imperative Program is established using Matlab(R2012a) for each parameters of equation (1.4). Using this program some intersting graphs are established. Also using this program reader may obtain large number of graphs of equation (1.4), which gives the eccentric charecteristics in the area of sequence of functions or class of polynomials.
5.1
Code of new sequence of functions:
function [Vn] = gnhypergeo(sigma,lambda,mu,alpha,a,k,s,x) %Graph of Vn(sigma,lambda,mu,alpha,a,k,s,x)\\ %V=Vn(sigma,lambda,mu,alpha,a,k,s,x)=(1/n!)*x^(-beta) %*hypergeom([sigma,lambda],mu,x.^k)*Tn^(a,s)(x^a*(s+x*D)(x^beta) %*hypergeom([sigma,lambda],mu,-x.^k)), where n=1,2,3,鈥syms x %n=input(’please enter n:’); n=1; W1= hypergeom([sigma,lambda],mu,-x.{^}k); y=(x.{^}alpha).*W1; for i=1:n y=(x.{^}a).*(s.*y+x.*diff(y)); end W2=hypergeom([sigma,lambda],mu,x.^k); v=(1./factorial(n)).*(1./(x.{^}alpha)).*W2.*y; Vn=subs(v,x); {end} Plot The Graph: hold on h1= ezplot(gnhypergeo(1,2,1,3,1,3,x),[-.1:.05:.1]); h2= ezplot(gnhypergeo(1,3,1,4,1,3,x),[-.1:.05:.1]); h3= ezplot(gnhypergeo(1,4,1,5,1,3,x),[-.1:.05:.1]); h4= ezplot(gnhypergeo(1,5,1,6,1,3,x),[-.1:.05:.1]); title(’V_1(1,lamba,1,alpha,1,3,x);lambda=2:5;alpha=3:6’);ylabel(’V_1’) xlabel(’x-axis’) hold off set(h1,’color’,’r’) set(h2,’color’,’b’) set(h3,’color’,’g’) set(h4,’color’,’k’) legend(’V_1(1,2,1,3,1,3,x)’,’V_1(1,3,1,4,1,3,x)’,’V_1(1,4,1,5,1,3,x)’, ’V_1(1,5,1,6,1,3,x)’) 207
P. Agarwal, et al : On new sequence of functions involving p Fq
5.2
Graphs
In this section, we establish here some graphs of new sequence of functions (1.4) with the help of using the above matlab program for different values of the parametrs. Figure 1, 2, 3 and 4 are established for the same parameters for n = 1, 2, 3, 4. Figure 5 is established for all the parameters of the sequence are same for n = 2 and rest figures are established for n = 1 and different values of the parameters. All the graphs can be easily interpreted, which are listed below as:
6
Conclusion
In this paper, we have presented a new sequence of functions involving the p Fq by using operational techniques.With the help of our main sequence formulae, some generating relations and finite summation formulae of the sequence are also presented here.Our sequence formulae is important due to presence of p Fq . On account of the most general nature of the p Fq a large number of sequences and polynomials involving simpler functions can be easily obtained as their special cases but due to lack of space we can not mention here. 208
South Asian J. Math. Vol. 3 No. 3
References 1 2 3 4 5 6
Chak, A. M., (1956), A class of polynomials and generalization of stirling numbers, Duke J. Math., 23, 45-55. Chandel, R.C.S., (1973), A new class of polynomials, Indian J. Math., 15(1), 41-49. Chandel, R.C.S., (1974), A further note on the class of polynomials Tnα,k (x, r, p), Indian J. Math.,16(1), 39-48. Chatterjea, S. K., (1964), On generalization of Laguerre polynomials, Rend. Mat. Univ. Padova, 34, 180-190. Gould, H. W. and Hopper, A. T., (1962), Operational formulas connected with two generalizations of Hermite polynomials, Duck Math. J., 29, 51-63. Joshi, C. M. and Prajapat, M. L., (1975), The operator Ta,k , and a generalization of certain classical polynomials,
209
P. Agarwal, et al : On new sequence of functions involving p Fq
7 8 9 10 11 12 13 14 15 16 17 18
210
Kyungpook Math. J., 15, 191-199. Mittal, H. B., (1971), A generalization of Laguerre polynomial, Publ. Math. Debrecen, 18, 53-58. Mittal, H. B., (1971), Operational representations for the generalized Laguerre polynomial, Glasnik Mat.Ser III, 26(6), 45-53. Mittal, H. B., (1977), Bilinear and Bilateral generating relations, American J. Math., 99, 23-45. Patil, K. R. and Thakare, N. K., (1975), Operational formulas for a function defined by a generalized Rodrigues formula-II, Sci. J. Shivaji Univ. 15, 1-10. Prajapati, J.C. and Ajudia, N.K., On sequence of functions and their MATLAB implementation, Internation journal of physical, Chemical and Mathematical sciences, Vol. No. 2 , ISSN:2278-683x, p.p. 24-34, Accepted On: 27.08.2012. Salembhai, I.A., Prajapti, J.C. and Shukla, A.K., On sequence of functions, Commun. Korean Math. 28(2013), No.1 ,p.p. 123-134. Shrivastava, P. N., (1974), Some operational formulas and generalized generating function, The Math. Education, 8, 19-22. Srivastava,H. M. and Choi,J.(2012), Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York. Shukla, A. K. and Prajapati J. C., (2007), On some properties of a class of Polynomials suggested by Mittal, Proyecciones J. Math., 26(2), 145-156. Singh, R. P., (1968), On generalized Truesdell polynomials, Rivista de Mathematica, 8, 345-353. Srivastava, A. N. and Singh, S. N., (1979), Some generating relations connected with a function defined by a Generalized Rodrigues formula, Indian J. Pure Appl. Math., 10(10), 1312-1317. Srivastava, H. M. and Singh, J. P., (1971), A class of polynomials defined by generalized, Rodrigues formula, Ann. Mat. Pura Appl., 90(4), 75-85.
