Further results on the family of generalizred radiation integrals

              Rev.Acad.Canar.Cienc., XI (Núms. 1·2), 71-81 (1999)
FURTHER RESULTS ON THE FAMILY OF
GENERALIZED RADIATION INTEGRALS
¡(p,q,u,v)
m,n
L. Galué
Centro de Investigación de Matemática Aplicada (C.I.M.A.)
Facultad de Ingeniería. Universidad del Zulia
Apartado 10482
Maracaibo-Venezuela
Abstract
In this paper we continue the investigation of the family of generalized ra­diation
integrals defined by
q
[
a, b, e, d, >. l 1-mp rr r([3j)
¡(p,q,u,v) a (a ) ([3 ) = <J a _1·=_1 __ _ m,n > P > q 4 p
(r¡u), bv) 7r r(a) rr r(aj)
X
j=l
{b x>-(xm+c)8Gn,p+1 (~ ¡ 1 + p - (ap), 1+8 ) F, ( (r¡u); xm) dx Jo p+l,q+l xm +e p, 1 + P - ([3q) u v bv); dm
where a, b, e, d > O; -1 < Re(>.) < mminRe(p, l+p-(f3j ))-m [maxRe(l - (r¡k)) - 1]­mRe(
8) - 1, j = 1,. . .,n - 1, k = 1,. . .,u; m,n,p,q E N; /i #- 0,-1, -2,. . .,
i=l,2, .. .,v.
For this family of generalized radiation integrals we present several cases of
reducing of the order, various recurrence relations and sorne relations of special
type.
Key words: Radiation integrals, recurrence relations, reducing of the order.
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Introduction
The response I(a, b) of an omnidirectional radiation detector at a height h
directly over a comer of a plane isotropic rectangular (plaque) so urce of length
l, width w and uniform strength a can be expressed as [3]:
I(a , b) = -a lb arctan (J X2a +1 ) JXd2x+ 1'
4n o x2 + 1 x2 + 1
(1)
where a= w/h >O, b = l/h >O.
This integral was originally investigated by Hubbell et al. [3] in civil de­fense
studies, in order to predict radiation fields due to radioactive fallout, from
nuclear accidents or weapons, deposited on rectangular surfaces such as roof
sections; the integral (1) has also subsequently found important applications in
irradiation technology.
Various generalizations of (1) have been given by several authors, so we have:
Kalla et al. [4] defined and studied the following integral involving the Gauss
hypergeometric function 2F1 (a, /3; /'; x) :
H [ a, b,p, >. ] = aa lb x>.(x2 +p)-" F (a f3· ,.,. _ _i!:_) dx "'/J'V 4 21 ''" 2+ ._.., ' 1 7r O X p
Re(!') > Re(/3) >O, -1 < >. < 2a - l;p, a, b >O.
Galué [1] generalized the integral (2) by means of
I [ ·a,b,p,>.,μ ]
a, /3, 1'
X
(2)
(3)
where Re(!')> Re(/3) >O; -1 < >. < 2a - 2μ-1; μ > -1; p,a,b >O; O< a:::;
b <OO.
Other generalization of (2) has been given by Saigo and Srivastava [6] in the
form
F [ a,(ap); am ] d
p+l q (/JqJ; - xm+c X
where min{a,b,c} >O, >. E (-1,1) and pFq((ap);(/3q);z)
hypergeometric function.
72
X
(4)
is the generalized
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Recently Galué and Prieto [2] have considered generalized radiation integrals
in the following forro:
q
] _ cr a _J=_l_ _
1-mp n r({3j)
- 47r fr r(aj)
X
j=l
rb x.X(xm + c)c5 Gn,p+l (~ 11 + p - (ap), 1+8 ) dx (5) Jo p+l,q+l xm +e p, 1 + p - ({3q)
a, b, e> O; - 1 <Re(>.)< m minRe(p, l+p-f3j)-mRe(8)-1, j = 1, ... , n-1;
m, n, p, q E N y G;'.:t ( z 1 ~: ) is the Meijer G-function,
and
q
[
a, b, e, d, >. l 1-mp n r({3j)
¡(p,q,u,v) a (a ) ({3) = cr a _J_·=_l _ _
m,n > P > q 4 p
(r¡u), bv) 7r r(a) n r(aj)
X
j=l
l+p-(ap),1+8) uFv( (r¡u): x:)dx
p,l+p-({3q) (/v), d
(6)
a, b, e, d >O; -1 <Re(>.) < mminRe(p, 1 + p - (f3j)) - m[maxRe(l - (r¡k)) -
1] - mRe(8) -1, j = 1, ... ,n -1, k = 1, ... ,u; m,n,p, q EN; /i =f. 0,-1,-2, ... ,
i = 1, 2, ... ,v.
In this paper we continue the investigation of the family of generalized ra­diation
integrals defined in (6), and we present several cases of reducing of the
order, various recurrence relations and sorne relations of special type.
Cases of Reducing of the Order
In this section, using the known formulas (7.2.3.14) through (7.2.3.21) in [5]
we establish many interesting results with positive integers h and i for reducing
the order of the integral (6). These are listed below:
¡(p,q,u,v) [ a (a ) ({3 ) = ¡(p,q,u-1,v-1) a (a ) ({3 )
a, b, e, d, >. l [ a, b, e, d, >.
m,n , P ' q m,n ' P ' q
( T/u- 1), /v + 1, bv) ( T/u- i), bv- 1)
u-1
jn=l T/j ~¡(p,q,u-1,v-1) [
v dm m,n n /j
j=l
a, b, e, d, >. + m l a, (ap), ({3q)
(T/u- i) + 1, bv-1) + 1
73
(7)
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[
a, b, e, d, >.
¡(p,q,u,v) a (a ) ({3 )
m,n ' P' q
(TJu-1),/v + h, bv)
[
a, b, e, d, >. l ¡(p,q,u-l,v-1) a (a ) ({3 )
m,n ' P' q ·
(TJu-1) + r, bv-1) + r
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ({3 ) =
m,n ' P' q
(rJu-2),/v-l +h,/v+i,(¡v)
[
a, b, e, d, >. l ¡(p,q,u-2,v-2) a (a ) ({3 )
m,n ' P' q
(TJu-2) + r + s, bv-2) + r + S
where we use the result [7, pg. 17, No. (12) and No. (9)]
O~k~n
k>n
(>.)m+n = (.A)m(A + m)n·
v-1
[
a, b, e, d, >. l n (¡j - 1) dm
I~;~,u,v) a, (ap), ({3q) = _1=_:_-1 _ _
(rJu-1), 1, bv-1), 2 n (r¡j - 1)
j=l
X
(8)
X
(9)
(10)
(11)
X
{ [
a, b, e, d, >. - m l [ b >. ] } I~;~,u- l,v-l) a, (ap), ({3q) . - !~;~) :', (~~), (~q) , (12)
(TJu-i) - 1, (lv-1) - 1
w1· t h T)1, .. .,rJu-14r 1 an d ¡(mp,,nq ) [ aa,, (ba, pe,)>,(. {3q) ] as d e fi ne d m· (5) .
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v-1
[
a, b, e, d, >. l f1 bi - 2)(¡i - 1) d2m
I (p ,q,u,v ) a (a ) ((3 ) = 2 J--1 X
m,n ' p ' q u-1
(7Ju-1), 1, bv-1), 3 n (7Jj - 2) (7Jj -1)
j=l
{ [
. a, b, e, d, >. - 2m l )I ( 1li - 2) 1 ¡(p,q,u- l ,v-1) a (a ) ((3 ) _____ _
m,n ' P ' q v-1 dm
(7Ju-l) - 2, bv-1) - 2 jgl (¡j - 2)
X
¡(p,q) [ a, b, e,>.+ m ] _ ¡(p,q) [ a, b, e,>. ] } .
m,n a , (ap), ((3q) m,n a, (ap), ((3q) (13)
7]1, · · · ·, 77u- l =J 1, 2; /1, · ... , /v-1 =J 2
v-1
[
a, b, e, d, >. l .IT (1 - 'Yi )h
¡(p,q,u,v) a (a ) ((3 ) = (-l)u-v J=l X
m,n ' P ' q u-1
(7Ju-1), 1, bv-1), h + 1 n (1 -17j)h
j=[
a, b, e, d, >. l
J~;~,u-1,v-l) a, (ap), ((3q) _
(1Ju-i) - h, bv-1) - h
v-1
h "'°""" 1.-D1 (1 - rj)s (-l)(u-v)(l+h-s) ¡(p,q) [ a, b, e,"'+ m (h - s ) ]
L., u-1 (h _ s)! dm(h-s) m,n a, (ap) , ((3q) '
s=l n (1 - 77j)s
(14)
j=l
where we have used the result [7, pg. 17, No. (10)]
O::::; k::::; n . (15)
¡(p,q,u,v) [ a (a ) ((3 ) = X
a, b, e, d, >. l 1
m,n ' P ' q ( )
(1Ju), bv-2), 77u-l + 1, 7]u + 1 7]u - 7]u-l
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{ [
a, b, e, d, >. l 7/u I!/:;~,u-1,v-l) a, (ap), ({Jq) -
(7/u-1), bv-2), 7/u-1 + 1
[
a, b, e, d, >.
r¡ 1 ¡(p,q,u-l,v-1) 0 (o ) ({3 )
u- m,n ' P ' q
(7/u-2),r¡u,(/v-2),r¡u+ 1
(16)
. [ a, b, e, d, >. l ( ) ¡(p,q,u,v) 0 (o ) ({3 ) 7/u-1 h
m,n ' P ' q - )
( ) ( ) + h + 1 7/u-1 - 7/u h
7/u ' /v-2 '7/u-1 , 7/u
X
[
a, b, e, d, >. l ( ) ¡(p,q,u-l,v-1) 0 (o ) ({3 ) _ 7/u-1 h7/u
m,n ' P ' q ( ) ( 7/u-2 ) ,r¡u, (/ v-2 ) ,r¡u + 1 7/u-1 - 7/u h
X
""h' "' ( 7/u-1 - 7/u - l)i ¡(p,q,u-1,v-l) [ 0a , b(,o e , )d , ({>3. ) l L_¿ ( ) m,n > P > q
i=l 7/u-l i (7/u-1), bv-2), 7/u-1 + i
(17)
7/u-1 =/= 7/u·
Recurrence Relations
In this section we establish various recurrence relations for I!/:;~·u,v) [· · ·] ,
where the upper parameters are different from the corresponding lower param­eters
by an integer. The results presented are based on the definition (6) and
formulas (7.2.3.25) through (7.2.3.36) in [5]. We list the following interesting
results without going into details.
7/u- 1 ¡(p,q,u,v) [ 0 (o ) ({3 )
a, b, e, d, >. l ( ) m,n ' P ' q
7/u - 7/u-l ( 7/u-2), 7/u-1 + 1, 7/u, (rv)
/v - 1 ¡(p,q,u,v)
(rv - 7/u - 1) m,n
76
[
a, b, e, d, >.
a, (ap), ({Jq)
(r¡u), bv-1), /v - 1
(18)
l -
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[
a, b, e, d, >. l 7/u ¡(p,q,u,v) a (a ) ((3 )
('y _ r¡ _ l) m,n ' P ' q ·
v u (7/u-1), 7/u + 1, bv)
(19)
[
a, b, e, d, >. l _ 1 [ a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) = "/v ¡(p,q,u,v) a (a ) ((3 ) _
m,n ' P ' q ("fv _ "/v-l) m,n ' P ' q
( 7/u)' bv) ( 7/u) , bv-1 ), "/v - 1
'Yv-1 - 1 ¡(p,q,u,v) [ aa, b(,a e , )d , ((>3. ) l ( ) m ,n > P > q ·
"/v -"(v-l (r¡u), bv-2),"fv-l -1,"fv
(20)
I (p,q,u,v) [ ( ) ((3 ) = 7/u-1 "/v - 7/u
a, b, e, d, >. l ( ) m,n a, ap , q ( ) x
( 7/u), ('rv) "/v 7/u-l - 7/u
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) -
m,n ' P' q
( 7/u-2), 7/u-1 + 1, 7/u, bv-1 ), "/v + 1
( )
[
a, b, e, d, >. l 7/u "/v - 7/u-1 ¡(p,q,u,v) a (a ) ((3 )
( ) m,n > P > q · "/v 7/u-1 - 7/u (7/u-1), 7/u + 1, bv-1), "/v + 1
(21)
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) -
m,n ' P' q
(r¡u), bv-2),"fv-l + l ,"fv - 1
( l) [ a,b,c,d,>. l 7/u 'Yv-1 - "/v + ¡(p,q,u,v) a (a ) ((3 )
"/v-1 ( "/v - 7/u - 1) m,n ' P ' q · (7/u-1), 7/u + 1, bv-2), "/v-1+1, "/v
(22)
[
a, b, e, d, >. l [ a, b, e, d, >.
¡(p,q,u,v) a (a ) ((3 ) = ¡(p,q,u,v) a (a ) ((3 )
m,n ' P ' q m,n ' P ' q
( 7/u)' bv) ( 7/u- 1), 7/u + 1, bv) l -
(23)
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'U
ul ( 7/j) 1 [ a, b, e, d, >. + m l J- ¡(p,q,u,v) a (a ) ((3 )
v-1 'Y(¡ +l)dm m ,n ' P' q ·
[I(¡j) v v (r¡u)+l,('Yv-l)+l,/v+2
j=l
¡(p,q,u,v) [ a (a ) ((3 ) = ¡(p,q,u,v) a (a ) ((3 )
a, b, e, d, >. l [ a, b, e, d, >.
m ,n ' P ' q m,n ' P ' q
(r¡u), ('Yv) (7/u-2), 7/u-1+1, 7/u - 1, ('Yv)
u-2
( 1) .rr ( 7/j) [ a, b, e, d, >. + m l 7/u - 7/u-1 - J=l ¡(p,q,u,v) a (a ) ((3 )
dm v m,n ' P ' q · rr (¡j) (71u- i) + 1,r¡u, (iv) + 1
j=l
[
a, b, e, d, >.
¡(p,q,u,v) a (a ) ((3 )
m,n ' P' q
( 7/u), (iv)
'U
l [ a, b, e, d, >.
= ¡(p,q,u,v) a (a ) ((3 )
m,n ' P' q
( 7/u-1 ), 7/u + 1, ('Yv-1), 'Yv + 1
rr ( 7/j) ( ) [ a, b, e, d, >. + m l J=l 'Yv - 7/u ¡(p,q,u,v) a (a ) ((3 )
vr-r1 'Y (¡ + l)dm m,n ' P ' q · (¡j) V V (r¡u) + 1, (iv-1) + l,"fv + 2
j=l
¡(p,q,u,v) a (a ) ((3 ) = 7/u-1 ¡(p,q,u,v) a (a ) ((3 )
(24)
]-
(25)
l -
(26)
[
a, b, e, d, >. l [ a, b, e, d, >.
m,n ' P ' q 'Y m,n ' P ' q
(r¡u), ('Yv) v (7/u-2),7/u-l + l,r¡u + 1, ('Yv-1),"fv + 1
(' Yv - 7/u-1 ) ¡(p,·q ,u,v) [ aa, b(,a e , )d , ((>3. ) - l m,n ' P' q
'Yv (7/u-1), 7/u + 1, ('Yv- 1), /v + 1
u-2 .rr ( 7/j) [ a, b, e, d, >. + m l J=l 7/u-1 ¡(p,q,u,v) a (a ) ((3 )
v dm m,n ' p ' q · rr (¡j) (r¡u) + 1, (iv) + 1
(27)
j=l
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I (p,q,u ,v) [ a, b(, c,)d,(>.(3 ) l = bv-2 - l)bv-1 - l)("lu-1 - /v )(r¡u - /v)
m,n a, ap > q ( ) X
(r¡u), (/v) 'f/u-1 - 1 (r¡u - 1)(/v-2 - /v)(/v-1 - /v)
¡(p,q,u,v) a (a ) ((3 )
m 1n ' P' q
[
a, b, e, d, >.
("lu-2),r¡u-l -1,r¡u -1, bv-3),/v-2 -1,/v-l -1,/v
bv-2 - 1)(/v - l)('T/u-1 - /v-i)(r¡u - /v-1)
X
('T/u-l - l)(r¡u - 1)(/v-2 - /v-1)(/v - /v-1)
¡(p,q,u,v) a (a ) ((3 )
m,n ' P' q
[
a, b, e, d, >.
('T/u-2),r¡u-l -1,r¡u -1, bv-3),/v-2 -1,/v-l,/v -1
bv-1 - 1)(/v - l)("lu-1 - /v-2)(r¡u - / v-2)
X
("lu-1 - l)(r¡u - 1)(/v-1 - /v-2)(/v - /v-2)
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) m,n > P > q ·
("lu-2),'f/u-l -1,r¡u - l,(¡v-2),/v-l -1,/v -1
I (p,q,u,v) [ ( ) ((3 ) = 'f/u-2 'f/u-1 /v-1 - 'f/u /v - 'f/u
a, b, e, d, >. l ( )( ) m,n a, ap , q ( )( ) X
( 'f/u), (/v) /v-1/v 'f/u-2 - 'f/u 'f/u-l - 'f/u
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) + m,n ' P' q
("lu-3), 'f/u-2 + 1, 'f/u-1+1, 'f/u, bv-2) , /v-1 + 1, /v + 1
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) + m,n ' P' q
('T/u-3), 'f/u-2+1, 'f/u-1, 'f/u + 1, bv-2), /v-1+1, /v + 1
¡(p,q,u,v) [ a (a ) ((3 )
a, b, e, d, >. l m n ' P' q ·
' ('T/u-2), 'f/u-1+1, 'f/u + 1, bv-2), /v-1+1, /v + 1
79
(28)
(29)
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Relations of Special Type
We list here four more recurrence relations for I$!:;~,u,v) [· · -J. From the defi­nition
(6) and formulas (7.2.3.38) to (7.2.3.41) in [5], it is easy to establish the
following relations that are independient of any multipliers.
[
a, b, e, d, >. l [ a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) +f(p,q,u,v) a (a ) ((3 )
m,n ' P ' q m,n ' P ' q
(r¡u), bv-1), 1- /v-1 (r¡u), bv-2), -/v-1,/v-l + 1
[
a,b,c,d,>. l = 2 ¡(p,q,u,v) a (a ) ((3 ) .
m,n ' P' q
(r¡u), bv-2), 1 - /v-1, 1 + /v-1
(30)
[
a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) =
m,n ' p' q
(r¡u), bv-2), -r¡u, 1+7/u
[
a, b, e, d, >. l 2 ¡(p,q,u,v) a (a ) ((3 ) -
m,n ' P' q
(r¡u), bv-2), 1 - 7/u, 1+7/u
[
a, b, e, d, >. l ¡(p,q,u-l,v-1) a (a ) ((3 )
m,n ' P' q ·
(7/u-1), bv-2), 1- 7/u
(31)
[
a, b, e, d, >. l I$!:;~,u,v) a, (ap) , ((3q) +
(r¡u) , bv-1), 1+7/u
[
a,b,c,d,>. l ¡(p,q,u,v) a (a ) ((3 )
m,n ' P' q
(7/u-i), -r¡u, bv-1), 1 - 7/u
[
a, b, e, d, >. l 2 I$;:;~·u+l,v+l) a, (ap), ((3q) .
(7/u-1), -r¡u, 7/u, bv-1), 1 - 7/ui 1+7/u
(32)
[
a, b, e, d, >. l [ a, b, e, d, >. l ¡(p,q,u,v) a (a ) ((3 ) + ¡(p,q,u,v) a (a ) ((3 )
m,n ' P ' q m,n ' P ' q
(7/u- 1), 1 - 7/u- li (/v) (7/u-2), -r¡u-1, 1+7/u-1, (/v)
= 2 ¡(p,q,u,v) [ a (a ) ((3 )
a, b, e, d, >. l m,n ' P' q ·
(7/u-2), -r¡u-li 7/u-1> bv)
(33)
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Acknowledgement
The author would like to thank CONDES - Universidad del Zulia for financial
support.
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81
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