In this paper, first we evaluate a finite integral involving general class of polynomials and the product of two H -functions and then we make its application to solve boundary value problem on heat conduction in a rod under the certain conditions and further we establish an expansion formula involving about product of H -function. In view of generality of the polynomials and products of H -function occurring here in, on specializing the coefficients of polynomials and parameters of the H -function, our results would readily reduce to a large number of results involving known class of polynomials and simpler functions.

1. International Journal of Technical Research and Applications e-ISSN: 2320-8163,
www.ijtra.com Volume 2, Issue 4 (july-aug 2014), PP. 122-126
122 | P a g e
H- FUNCTION AND GENERAL CLASS OF
POLYNOMIAL AND HEAT
CONDUCTION IN A ROD.
Dr. Rachna Bhargava
Department of Mathematics
Global College of Technology (GCT), Jaipur-302022, India
Abstract - In this paper, first we evaluate a finite
integral involving general class of polynomials and the
product of two H -functions and then we make its
application to solve boundary value problem on heat
conduction in a rod under the certain conditions and
further we establish an expansion formula involving
about product of H -function. In view of generality of
the polynomials and products of H -function occurring
here in, on specializing the coefficients of polynomials
and parameters of the H -function, our results would
readily reduce to a large number of results involving
known class of polynomials and simpler functions.
Keywords: General Class of Polynomials, H Function,
Jacobi polynomial and Leguerre polynomials.
Mathematics Subject Classification : 33C60, 34B05
I. INTRODUCTION
The general class of polynomials introduced by
Shrivastava [7] and defined by [8] and [10] as follows:
0,1,2,...nx
!k
An)
x]S kkn,km,
[n/m]
0k
m
n


 

….. (1)
where m is an arbitrary positive integer the coefficient
An,k (n,k ≥ 0) are arbitrary constants, real or complex.
H -function will be defined and represented as
follows [2] and [4]:






 

P1,NjjaN1,jAjj(a
Q1,MjBjjbM1,jjb
NM,
QP,
NM,
QP,
zHz]H


 


 dz
i2
1 i
i
…(2) where  ≠ 0
and







jj
P
1Nj
jB
jj
Q
1Mj
jA
jj
N
1j
jj
M
1j
ab1
a1b
…(3)
and also the H -function occurring in the paper was
introduced by Inayat-Hussain [4] and studied by
Bushman and Shrivastava [2]. The following series
representation for the H -function was obtained by
Rathie [5].






 

P1,NjjcN1,jCjj(c
Q1,MjDjjdM1,jjd
NM,
QP,
NM,
QP,
zHz]H
rh,
h
r
rh,jj
P
1j
jD
rh,jj
Q
1Mj
jC
rh,jj
N
1j
rh,jj
M
hj
1jM
1h0r
z
!r
1
cd1
c1d







 







r
r
rh,
hd


 …(4)
The nature of contour L and series of various conditions
on its parameters can be seen in the paper by Bushman
and Shrivastava [2]. We shall also make use of the
following behaviour of the z]H
NM,
QP,
 function for
small value of f(z) as recorded by Saxena [6, p.112,
eq.(2.3) and (2.4)]
 
|z|(0z}H
NM,
QP,
for small z
where (2)for)/(dRemin jj
Mj1



and (4)for)/(bRemin jj
Mj1



The following more general conditions given by
0TT
2
1
(zarg 111

and 0TT
2
1
(zarg 222
 .
where
0BAT j
1P
11Nj
jj
1Q
11Mj
jj
1N
1j
j
1M
1j
1
 

and
0DCT j
2P
12Nj
jj
2Q
12Mj
jj
2N
1j
j
2M
1j
2
 

.

2. 123 | P a g e
II. MAIN INTEGRAL













 













 





 


2b
2
2m
2n
1b
1
1m
1n
m
1uL
0 L
x
sinyS
L
x
sinyS
L
x
sin
L
x
sin













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH
dx
L
x
sinzH
2P12Njjc
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P 












  



1k
1
1
1k1n1k1m1rh,2h2k2b1k1b2M
12h0r
2m2n
02k
1m1[n
01k
y
k
An
2k












2
siny
k
An
m2k
2
2
2k2n2k2m2 

 
!r
z1
cd1
c1d
h
rh,
2
r
rh,jj
2P
12Nj
jD
rh,jj
2Q
12Mj
jC
rh,jj
2N
1j
rh,jj
2M
hj
1j


















 



1P11Njja
1N1,jAjja11h1rh,2h2k2b1k1bu
11hmrh,2h2k2b1k1bu
1Q1,1MjBjjb
1M1,jj(bh
111N1M
11Q11P
2
z
H

…(5)
Where (i)
1u
2Lk 

(ii) 0kandknnhh 212121
 and
provided that conditions
(i)
h
h
j
j rb
and0
b
Remin












(ii)
h
h
rh,
j
j rd
and0
d
Remin












(iii)
0TBA 1j
1P
11Nj
jj
1Q
11Mj
jj
1N
1j
j
1M
1j
 

where  11
T
2
1
zarg
(iv)
0qDC 2j
2P
12Nj
jj
2Q
12Mj
jj
2N
1j
j
2M
1j
 

where  22
T
2
1
zarg
(v) 0hhkbkb{uRe 1rh,22211

Proof : To establish the above integral (5), we first express
both the general class of polynomials and
z]H
2N2M
2Q2p



occurring in its left hand side in their
respective series forms with the help of equation (1) and (3)
respectively and then interchange the order of integration
and summation (which is permissible under the condition
stated) and using (3) and with the help of x-integral given by
Gradshteyn, I.S. and Ryzhik [3]. Then we substitute the
above  with the help of (4) and reinterpret the result
thus in terms of H -function, we arrive at the right hand
side of desired results (5).
III. MAIN PROBLEM
Problem of heat conduction in a rod with one end
held at zero temperature and the other end exchanges heat
freely with the surrounding medium at zero temperature. If
the thermal coefficients are constants and there are no
source of thermal energy, then temperature in a one-
dimensional rod 0  x  L satisfies the following heat
equation
0t
t
k
t 2
2






…(6)
In view of the problem, the solution of this partial
differential equation satisfy the boundary conditions
0t)0  …(7)
0t)L,ht)L,
t



…(8)
 (x,t) is finite as t   …(9)
The initial condition
 (x,0) = f(x) …(10)
The solution of partial differential equation (6) can be
written as [11, p.77,(4)]



















 



kt
2
L
m
mm
1m
e
L
x
sinBt)x,
…(11)
at t = 0













 













 





 

 2b
2
2m
2n
1b
1
1m
1n
1u
L
x
sinyS
L
x
sinyS
L
x
sinf(x)x,0)













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH













  



2P12Njjc
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P L
x
sinzH
…(12)
The solution of the problem to be obtained is








 rh,2b2k2b1k1b2m
12h0r
2m2n
02k
1m1[n
01k
2k't)x,

3. 124 | P a g e
2k
2
2
2k2n2k2m2
1k
1
1
1k1n1k1m1
y
k
An
y
!k
An

 
!r
z)1(
)c()}d1({
)}c1({)d(
h
2
r
rh,jj
P
1Nj
D
rh,jj
Q
1Mj
C
rh,jj
N
1j
rh,jj
M
hj
1j rh,
2
2
j
2
2
j
22




























 





 









 

kt
L
exp
L
x
sin
2sin2
2
sin 2
m
m
mm
m
m







 



1P11Njja
1N1,jAjja11h1rh,2h2k2b1k1bu
11hmrh,2h2k2b1k1bu
1Q1,MjBjjb
1M1,jj(bh
111N1M
11Q11P
2
z
H

…(13)
Where L2
d
4
k' 1u



valid under the condition of (5).
Proof: The solution of the problem stated is given as
[11, p.77, (4)]



















 



kt
2
L
m
mm
1m
e
L
x
sinBt)x, where
1, 2,… are rots of transcendental equation.
kL
tan m
m


If t = 0, then by virtue of (11) and (12), we have













 













 





 
 2b
2
2m
2n
1b
1
1m
1n
1u
L
x
sinyS
L
x
sinyS
L
x
sin













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH













  



2P12Njjc
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P L
x
sinzH
L
x
sinB m
m
1m

 


…(13)
Multiplying both sides of (13) by
L
x
sin m

 and
integrate with respect to x from 0 to L, we get













 













 





 


2b
2
2m
2n
1b
1
1m
1nm
1uL
0 L
x
sinyS
L
x
sinyS
L
x
sin
L
x
sin













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH













  



2P12Njjc
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P L
x
sinzH
dx
L
x
sin
L
x
sinB mm
L
0
m
1m



 


…(14)
and using (5) and orthogonality property [12, p.28] by
Szego, we obtain
mm
m
21
m
m 2sin2
2
sin
d
4
B








 





 











 rh,2b2k2b1k1b2m
12h0r
2m2n
02k
1m1[n
01k
2k'
2
siny
k
An
y
!k
An
m2k
2
2
2k2n2k2m2
1k
1
1
1k1n1k1m1 

 
!r
z1
cd1
c1d
h
rh,
2
r
rh,jj
2P
12Nj
jD
rh,jj
2Q
12Mj
jC
rh,jj
2N
1j
rh,jj
2M
hj
1j


























 




1P11Njja
1N1,jAjja11h1rh,2h2k2b1k1bu
11hmrh,2h2k2b1k1bu
1Q1,MjBjjb
1M1,jj(bh
111N1M
11Q11P
2
z
H

…(15)
with the help of (14) and (11), we arrive at the right hand
side of desired result.
IV. EXPANSION FORMULA













 













 





 
 2b
2
2m
2n
1b
1
1m
1n
1u
L
x
sinyS
L
x
sinyS
L
x
sin













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH













  



2P12Njjc
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P L
x
sinzH








 rh,2b2k2b1k1b2m
12h0r
2m2n
02k
1m1[n
01k
2k'
2k
2
2
2k2n2k2m2
1k
1
1
1k1n1k1m1
y
k
An
y
!k
An

 
!r
z1
cd1
c1d
h
rh,
2
r
rh,jj
2P
12Nj
jD
rh,jj
2Q
12Mj
jC
rh,jj
2N
1j
rh,jj
2M
hj
1j












4. 125 | P a g e







 







 

L
x
sin
2
sin mm
m















 




1P11Njja
1N1,jAjja11h1rh,2h2k2b1k1bu
11hmrh,2h2k2b1k1bu
1Q1,MjBjjb
1M1,jj(bh
111N1M
11Q11P
2
z
H

…(16)
where all the conditions of (5) are satisfied.
Proof: Using (12) and (15) in (11), we arrive at the
expansion formula
14
2L
d
4
k' 



6. SPECIAL CASES OF (12)
(i) If
k
kn, 1
1)n
n
n
A







 

then 2x)1Px]s
(
n
'
n


where x)P
(
n


is Jacobi polynomial [10, p.68, eq.
(4.3.2)] and also
k
kn, 1
1
n
n
A1,m





 

then x)Lx]S (
Ln
'
n
 
where

Ln
L (x) is Leguerre Polynomial [10, p.101, eq.
(5.1.6)] and we get





 





 





 



L
x
sinL
L
x
sin21P
L
x
sinf(x)
2n1n
1u













  



1P11Njja
1N1,jAjja
1Q11MjBjjb
1M1,jj(b
1h
1
1N1M
1Q1P L
x
sinzH













  



2P12Njjd
2N1,jCjjc
2Q12MjDjjd
2M1,jj(d
2h
2
2N2M
2Q2P L
x
sinzH








 rh,2b2k2b1k1b2M
12h0r
2m2n
02k
1m1[n
01k
2k' .
2k
2
2
2k2m2
1k
1
1
1k1m1
y
k
n
y
!k
n

 















 







 








 

kt
L
exp
L
x
sin
2sin2
2
sin 2
mm
mm
m
m
!r
z1
cd1
c1d
h
rh,
2
r
rh,jj
2P
12Nj
jD
rh,jj
2Q
12Mj
jC
rh,jj
2N
1j
rh,jj
2M
hj
1j



















 

















 
1
1
n
n
1
1n
n
n
2
2
1k
1k1
1
1















 




1P11Njja
1N1,jAjja11h1rh,2h2k2b1k1bu
11hmrh,2h2k2b1k1bu
1Q1,MjBjjb
1M1,jj(bh
111N1M
11Q11P
2
z
H

(ii) If we substitute
 1k(0,A1k(0,A0nn0mm 212121
1N1M
1Q1Pjj2222221
H,1BAand0h1QM0PN0nn



then it reduces to Fox H-function and
Q)1,...,j;P1,...,i(Cjj
 it reduces
to the well known Meijer’s G-function by [9] in (12)
then we get a known result given in [1].
On applying the same procedure as above in
(16) , then we can establish the other known results.
ACKNOWLEDGEMENT
The authors are thankful to refree for his
valuable suggestions.
REFERENCES
1. Bajpai, S.D. and Mishra, Sadhana; (1991)
Meijer’s G-function and Fox’s H-function and
Heat conduction in a rod under typical
boundary conditions, Jnanabha, Vol.21 .
2. Bushman, R.G. and Shrivastava, H.M. (1990)
The H -function associated with a certain
class of Feynman integral. J. Phys. A: Math.
Gen. 23, 4707-4710 .
3. Gradshteyn, I.S. and Ryzhik, I.M. (1980)
Tables of Integrals, Series and Products,
Academic Press, Inc. New York .
4. Inayat-Hussain,,A.A.(1987) New Properties of
Hypergeometric series derivable from
Feynman integrals: II A generalization of the
H-function, J. Phys. A: Math. Gen. 20 , 4119-
4128.
5. Rathie,A.K.(1997) A new generalization of
generalized hypergeometric function, Le
Matematiche Fasc II, 52 , 297-310.
6. Saxena, R.K., Chena, Ram and Kalla, S.L.
(2002) Application of generalized H-function
in vibariate distribution, Rev. Acad. Can.
Ci.enc. XIV (Nums 1-2) , 111-120.
7. Shrivastava, H.M. (1972) A contour integral
involving Fox’s H-function, Indian J. Math. 14
, 1-6.
8. Shrivastava, H.M. (1983) The Weyl fractional
integral of a general class of polynomials, Bull.
Un. Math. Ital. (6) 2B , 219-228.

5. 126 | P a g e
9. Shrivastava, H.M., Gupta, K.C. and Goyal,
S.P. (1982) The H-Function of One and Two
Variables with Applications, South Asian
Publishers, New Delhi and Madras .
10. Shrivastava, H.M. and Singh, N.P. (1983) The
integration of certain products of the
multivariable H-function with a general class
of polynomials, Rend. Circ. Mat. Palermo
2(32) , 157-187.
11. Sommerfeld, A. (1949) Partial Differential
Equations in Physics, Academic Press, New
York .
12. Szego, G. (1975) Orthogonal polynomials.
(Amer. Math. Soc. Collog. Publ. Vol.23), 4th
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