Certain integrals involving ℵ-functions and Laguerre polynomials
UDC 517.5 Our aim is to establish certain new integral formulas involving $\aleph$ -functions associated with Laguerre-type polynomials. We also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler known functions, e.g., the generalize...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507303318388736 |
|---|---|
| author | Agarwal, P. Chand, M. Choi, J. Агарвал, Р. П. Чанд, М. Чой, Дж. |
| author_facet | Agarwal, P. Chand, M. Choi, J. Агарвал, Р. П. Чанд, М. Чой, Дж. |
| author_sort | Agarwal, P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:57:49Z |
| description | UDC 517.5
Our aim is to establish certain new integral formulas involving $\aleph$ -functions associated with Laguerre-type polynomials. We
also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler
known functions, e.g., the generalized hypergeometric function $_pF_q$ in a fairly systematic way. |
| first_indexed | 2026-03-24T02:07:10Z |
| format | Article |
| fulltext |
UDC 517.5
P. Agarwal (Anand International College Engineering, Jaipur, India),
M. Chand (Fateh College for Women, Bathinda, India),
J. Choi (Dongguk Univ., Gyeongju, Korea)
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS
AND LAGUERRE POLYNOMIALS
ДЕЯКI IНТЕГРАЛИ, ЩО ВКЛЮЧАЮТЬ \aleph -ФУНКЦIЇ
ТА ПОЛIНОМИ ЛАГЕРРА
Our aim is to establish certain new integral formulas involving \aleph -functions associated with Laguerre-type polynomials. We
also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler
known functions, e.g., the generalized hypergeometric function pFq in a fairly systematic way.
Наша мета — встановити деякi новi iнтегральнi формули, що включають \aleph -функцiї, асоцiйованi з полiномами
лагеррiвського типу. Також показано, що основнi результати, отриманi у статтi, є загальними. Для цього наведено
18 iнтегральних формул, що включають бiльш простi вiдомi функцiї, наприклад узагальнену гiпергеометричну
функцiю pFq в досить загальному виглядi.
1. Introduction and preliminaries. Let \BbbC , \BbbR , \BbbR +, \BbbZ and \BbbN be sets of complex numbers, real and
positive real numbers, integers and positive integers, respectively, and
\BbbZ -
0 := \BbbZ \setminus \BbbN , \BbbN 0 := \BbbN \cup \{ 0\} .
The Aleph (\aleph )-function, which is a very general higher transcendental function and was intro-
duced by Südland et al. [15, 16], is defined by means of Mellin – Barnes type integral in the following
manner (see, e.g., [8, 9]):
\aleph [z] = \aleph m,n
pk,qk,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (aj , Aj)1,n, [\delta j(ajk , Ajk)]n+1,pk;r
(bj , Bj)1,m, [\delta j(bjk , Bjk)]m+1,qk;r
\Biggr]
=
=
1
2\pi i
\int
L
\Omega m,n
pk,qk,\delta k;r
(s)z - s ds, (1.1)
where z \in \BbbC \setminus \{ 0\} , i =
\surd
- 1 and
\Omega m,n
pk,qk,\delta k;r
(s) :=
\prod m
j=1 \Gamma (bj +Bjs)
\prod n
j=1 \Gamma (1 - aj - Ajs)\sum r
k=1 \delta k
\prod qk
j=m+1 \Gamma (1 - bjk - Bjks)
\prod pk
j=n+1 \Gamma (ajk +Ajks)
. (1.2)
Here \Gamma is the familiar Gamma function (see, e.g., [13], Section 1.1); the integration path L = Li\gamma \infty ,
\gamma \in \BbbR , extends from \gamma - i\infty to \gamma + i\infty with indentations, if necessary; the poles of the Gamma
function \Gamma (1 - aj - Ajs), j, n \in \BbbN , 1 \leq j \leq n, do not coincide with those of \Gamma (bj+Bjs), j,m \in \BbbN ,
1 \leq j \leq m; the parameters pk, qk \in \BbbN satisfy the conditions 0 \leq n \leq pk, 1 \leq m \leq qk, \delta k \in \BbbR +,
1 \leq k \leq r; the parameters Aj , Bj , Ajk , Bjk \in \BbbR + and aj , bj , ajk , bjk \in \BbbC ; the empty product in
c\bigcirc P. AGARWAL, M. CHAND, J. CHOI, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9 1159
1160 P. AGARWAL, M. CHAND, J. CHOI
(1.2) (and elsewhere) is (as usual) understood to be unity. The existence conditions for the defining
integral (1.1) are given as follows:
\varphi \ell \in \BbbR + and | \mathrm{a}\mathrm{r}\mathrm{g}(z)| < \pi
2
\varphi \ell , \ell \in \BbbN , 1 \leq \ell \leq r,
and
\varphi \ell \geq 0, | \mathrm{a}\mathrm{r}\mathrm{g}(z)| < \pi
2
\varphi \ell and \Re (\varsigma \ell ) + 1 < 0,
where
\varphi \ell :=
n\sum
j=1
Aj +
m\sum
j=1
Bj - \delta \ell
\left( p\ell \sum
j=n+1
Aj\ell +
q\ell \sum
j=m+1
Bj\ell
\right)
and
\varsigma \ell :=
n\sum
j=1
Aj +
m\sum
j=1
Bj - \delta \ell
\left( p\ell \sum
j=n+1
Aj\ell +
q\ell \sum
j=m+1
Bj\ell
\right) +
1
2
(p\ell - q\ell ),
\ell \in \BbbN , 1 \leq \ell \leq r.
Remark 1. The expression in (1.1) of the Aleph-function does not follow completely the nota-
tional convention of the Fox’s H -function. Namely, in the \aleph -functions, the kernel \Omega m,n
pk,qk,\delta k;r
(s),
parameter couples
\bigl(
aj , Aj
\bigr)
1,n
,
\bigl(
bj , Bj
\bigr)
1,m
build the Gamma function terms exclusively in the nu-
merator, and
\bigl[
\delta j (ajk , Ajk)n+1,pk
\bigr]
,
\bigl[
\delta j
\bigl(
bjk , Bjk
\bigr)
n+1,qk
\bigr]
build the linear combination exclusively in
the denominator, while, for the Hm,n
p,q [z], both upper and lower couples of parameters
\bigl(
aj , Aj
\bigr)
1,p
and
\bigl(
bj , Bj
\bigr)
1,q
play roles in forming both numerator and denominator terms according to m and n.
Remark 2. Setting \delta j = 1, j \in \BbbN , 1 \leq j \leq r, in (1.1) yields the I -function (see [7]) whose
Further, special case when r = 1 reduces to a familiar function (see [3, 4]).
Prabhaker and Suman [5] defined the following Laguerre-type polynomials L
(\alpha ,\beta )
n (x) as follows:
L(\alpha ,\beta )
n (x) =
\Gamma (\alpha n+ \beta + 1)
n!
n\sum
k=0
( - n)k x
k
k! \Gamma (\alpha k + \beta + 1)
, (1.3)
n \in \BbbN , \Re (\alpha ) > 0, \Re (\beta ) > - 1,
where (\lambda )n is the Pochhammer symbol defined (for \lambda \in \BbbC ) by
(\lambda )n :=
\Biggl\{
1, n = 0,
\lambda (\lambda + 1) . . . (\lambda + n - 1), n \in \BbbN ,
=
=
\Gamma (\lambda + n)
\Gamma (\lambda )
, \lambda \in \BbbC \setminus \BbbZ -
0 .
The special case of (1.3) when \alpha = 1 reduces to the familiar generalized Laguerre polynomials L(\beta )
n (x)
(see, e.g., [6], Chapter 12):
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1161
L(1,\beta )
n (x) =
\Gamma (n+ \beta + 1)
n!
n\sum
k=0
( - n)kx
k
k! \Gamma (k + \beta + 1)
= L(\beta )
n (x).
The Konhauser polynomials of the second kind (see [12]) is defined by
Z\beta
n (x; k) =
\Gamma (kn+ \beta + 1)
n!
n\sum
j=0
( - 1)j
\Biggl(
cn
j
\Biggr)
xkj
\Gamma (kj + \beta + 1)
, (1.4)
\Re (\beta ) > - 1, k \in \BbbZ , n \in \BbbN .
It is easy to see that
L(0,\beta )
n (xk) = Z\beta
n (x; k)
and
L(\beta )
n (x) = Z\beta
n (x; 1). (1.5)
The polynomials Z
(\alpha ,\beta )
n (x; k) are defined as follows (see [10]):
Z(\alpha ,\beta )
n (x; k) =
n\sum
j=0
\Gamma (kn+ \beta + 1)( - 1)jxkj
j!\Gamma (kj + \beta + 1)\Gamma (\alpha n - \alpha j + 1)
, (1.6)
\Re (\alpha ) > 0, \Re (\beta ) > - 1, n \in \BbbN , k \in \BbbZ .
We find from (1.4) and (1.6) that
Z(1,\beta )
n (x; k) = Z\beta
n (x; k).
When \alpha \in \BbbN , (1.6) can be written in the following form:
Z(\alpha ,\beta )
n (x; k) =
\Gamma (kn+ \beta + 1)
\Gamma (\alpha n+ 1)
n\sum
m=0
( - \alpha n)\alpha mxkm
m!\Gamma (km+ \beta + 1)( - 1)(\alpha - 1)m
.
The polynomials L
(\alpha ,\beta )
n (\gamma ;x) are defined by (see [10])
L(\alpha ,\beta )
n (\gamma ;x) =
n\sum
r=0
\Gamma (\alpha n+ \beta + 1)( - 1)rxr
r!\Gamma (\alpha r + \beta + 1)\Gamma (\gamma n - \gamma r + 1)
,
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\alpha ), \Re (\gamma )\} > 0, \Re (\beta ) > - 1, n \in \BbbN .
Also we recall some properties of the Pochhammer symbol (see, e.g., [11])
( - x)n = ( - 1)n(x - n+ 1)n, (1.7)
(x+ y)n =
n\sum
j=0
\biggl(
n
j
\biggr)
(x)j(y)n - j , (1.8)
(x)n+m = (x)n(x+ n)m, (1.9)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1162 P. AGARWAL, M. CHAND, J. CHOI\biggl(
x
n
\biggr)
=
( - 1)n
n!
( - x)n, (1.10)
where x, y \in \BbbC and m, n \in \BbbN 0.
Here, in this paper, we aim to establish certain new integral formulas involving \aleph -function
associated with the Laguerre-type polynomials. We also show how the main results presented here
are general by choosing to demonstrate 18 integral formulas involving simpler known and familiar
functions, for example, the generalized hypergeometric function pFq, in a rather systematic manner.
2. Integral formulas. Here we present certain integral formulas mainly involving
the \aleph -functions.
Theorem 1. Let z, \lambda , \delta \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let C \in \BbbR + and
\Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the
following integral formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1\aleph m,n
\rho k,\sigma k,\delta k;r
\bigl[
zu - C
\bigr]
du =
= \Gamma (\delta )\aleph m,n+1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta , C), (aj , Aj)1,n, [\delta j(ajk , Ajk)]n+1,\rho k;r
(bj , Bj)1,m, (\lambda ,C), [\delta j(bjk , Bjk)]m+1,\sigma k;r
\Biggr]
(2.1)
provided the other involved parameters are so constrained that each member can exist.
Proof. Let \scrL 1 be the left-hand side of (2.1). Then, using (1.1) and changing the order of the
double integrals, which is guaranteed under the given conditions, we obtain
\scrL 1 =
1
2\pi i
\int
L
\Omega m,n
\rho k,\sigma k,\delta k;r
(s)z - s
\left\{
1\int
0
u\lambda +Cs - 1(1 - u)\delta - 1 du
\right\} ds. (2.2)
Recall the familiar Beta function B(x, y) which is defined by and expressed in terms of the Gamma
function as follows (see, e.g., [13, p. 8]):
B(x, y) =
\left\{
\int 1
0
tx - 1(1 - t)y - 1 dt, \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (x), \Re (y)\} > 0,
\Gamma (x) \Gamma (y)
\Gamma (x+ y)
, x, y \in \BbbC \setminus \BbbZ -
0 ,
to evaluate the inner integral in (2.2) to yield
\scrL 1 = \Gamma (\delta )
1
2\pi i
\int
L
\Omega m,n
\rho k,\sigma k,\delta k;r
(s)z - s \Gamma (\lambda + Cs)
\Gamma (\lambda + \delta + Cs)
ds. (2.3)
Finally, interpreting the right-hand side of (2.3) in terms of the definition (1.1), we arrive at the
right-hand side of (2.1).
Theorem 2. Let z, \lambda , \delta \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let x, t \in \BbbR with
x \geq t. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the
integration path Li\gamma \infty in (1.1). Then the following integral formula holds true:
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1163
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1\aleph m,n
\rho k,\sigma k,\delta k;r
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1\times
\times \aleph m,n+1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta , C), (aj , Aj)1,n, [\delta j(ajk , Ajk)]n+1,\rho k;r
(bj , Bj)1,m, (\lambda ,C), [\delta j(bjk , Bjk)]m+1,\sigma k;r
\Biggr]
(2.4)
provided the other involved parameters are so constrained that each member can exist.
Proof. Let \scrL 2 be the left-hand side of (2.4) and change the variable u into v =
u - t
x - t
. Similarly
as in the proof of Theorem 1, we can obtain
\scrL 2 =
(x - t)\delta +\lambda - 1
2\pi i
\int
L
\Omega m,n
\rho k,\sigma k,\delta k;r
(s)z - s(x - t)Cs
\left\{
1\int
0
(1 - v)\delta - 1v\lambda +Cs - 1 dv
\right\} ds =
= \Gamma (\delta )(x - t)\delta +\lambda - 1 1
2\pi i
\int
L
\Omega m,n
\rho k,\sigma k,\delta k;r
(s)z - s(x - t)Cs \Gamma (\lambda + Cs)
\Gamma (\delta + \lambda + Cs)
ds.
Now it is easy to see that, in view of the definition (1.1), the last equality is interpreted into the
right-hand side of (2.4).
Theorem 3. Let z, \nu , \mu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\nu ), \Re (\mu )\} > 0 and x \in \BbbR +. Also let C \in \BbbR + and
\Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the
following integral formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1\aleph m,n
\rho k,\sigma k,\delta k;r
\bigl[
z(x - t) - C
\bigr]
dt =
= x\nu +\mu - 1\Gamma (\nu )\aleph m,n+1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , C), (aj , Aj)1,n, [\delta j(ajk , Ajk)]n+1,\rho k;r
(bj , Bj)1,m, (\mu ,C), [\delta j(bjk , Bjk)]m+1,\sigma k;r
\Biggr]
(2.5)
provided the other involved parameters are so constrained that each member can exist.
Proof. A similar argument as in the proof of either Theorem 1 or Theorem 2 can establish the
result (2.5). So we choose to omit the details of its proof.
For the sequel of the above theorems, we need the following formula (see [1]) which is recalled
in Lemma 1.
Lemma 1. Let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a), \Re (c), \Re (\zeta ), \Re (\xi )\} > 0, \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b), \Re (d)\} > - 1 and h, m, n \in \BbbN .
Then the following formula holds true:
L(a,b)
n (\xi ;x)L(c,d)
m (\zeta ;x) =
m+n\sum
h=0
h\sum
k=0
\Gamma (an+ b+ 1)\Gamma (cm+ d+ 1)
\Gamma (h - k + 1)\Gamma (\zeta (m - h+ k) + 1)\Gamma (k + 1)
\times
\times ( - x)h
\Gamma (\xi (n - k) + 1)\Gamma (ak + b+ 1)\Gamma (c(h - k) + d+ 1)
. (2.6)
Theorem 4. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1164 P. AGARWAL, M. CHAND, J. CHOI
where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the following
formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (1 - u))L(c\prime ,d\prime )
n (\xi ;\sigma (1 - u))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
zu - C
\bigr]
du =
m+n\sum
h=0
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \Gamma (\delta + h)\sigma h\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda ,C)[\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
(2.7)
provided the other involved parameters are so constrained that each member can exist. Here
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given by
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime :=
h\sum
k=0
\biggl(
h
k
\biggr)
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)( - 1)h
\Gamma (\zeta (m - h+ k) + 1)\Gamma (\xi (n - k) + 1)
\times
\times 1
\Gamma (h+ 1)\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k\prime ) + d\prime + 1)
. (2.8)
Proof. Let \scrL 3 be the left-hand side of (2.7). Then, by using (2.6), we have
\scrL 3 =
m+n\sum
h=0
h\sum
k=0
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)(\sigma )h
\Gamma (h - k + 1)\Gamma (\zeta (m - h+ k) + 1)\Gamma (k + 1)
\times
\times ( - 1)h
\Gamma (\xi (n - k) + 1)\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\times
\times
1\int
0
u\lambda - 1(1 - u)\delta +h - 1\aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
zu - C
\bigr]
du. (2.9)
Applying (2.1) to the integral in (2.9), we obtain
\scrL 3 =
m+n\sum
h=0
h\sum
k=0
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)(\sigma )h
\Gamma (h - k + 1)\Gamma (\zeta (m - h+ k) + 1)\Gamma (k + 1)
\times
\times ( - 1)h
\Gamma (\xi (n - k) + 1)\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\Gamma (\delta + h)\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda ,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
. (2.10)
Finally, it is easy to see that the expression in (2.10) corresponds with the right-hand side of (2.7).
Here we present five integral formulas asserted in Theorems 5 – 9, without their proofs, because
each of their proofs would run parallel to that of Theorem 4.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1165
Theorem 5. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let x, t \in \BbbR with x \geq t, C \in \BbbR +
and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then
the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (u - t))L(c\prime ,d\prime )
n (\xi ;\sigma (u - t))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1
m+n\sum
h=0
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \sigma
h\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda + h,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist. Here
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given as in (2.8).
Theorem 6. Let z, \mu , \nu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let x \in \BbbR +, C \in \BbbR +, and
\Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the
following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (x - t))L(c\prime ,d\prime )
n (\xi ;\sigma (x - t))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(x - t) - C
\bigr]
dt = x\mu +\nu - 1
m+n\sum
h=0
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \sigma
h xh\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\mu ,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist. Here
\Delta n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given as in (2.8).
Theorem 7. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma
where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the following
formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (1 - u))L(c\prime ,d\prime )
n (\xi ;\sigma (1 - u))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
zu - C
\bigr]
du =
m+n\sum
h=0
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \Gamma (\delta + h)\sigma h\times
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1166 P. AGARWAL, M. CHAND, J. CHOI
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda ,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
(2.11)
provided the other involved parameters are so constrained that each member can exist. Here the
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given by
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime :=
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
\Gamma (\zeta m+ 1)\Gamma (\xi n+ 1)
\times
\times
h\sum
k=0
\Biggl[ \biggl(
h
k
\biggr)
( - 1)h - \zeta (h - k) - \xi k( - \zeta m)\zeta (h - k)( - \xi n)\xi k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\Biggr]
. (2.12)
Theorem 8. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let x, t \in \BbbR with x \geq t, C \in \BbbR +,
and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then
the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (u - t))L(c\prime ,d\prime )
n (\xi ;\sigma (u - t))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1
m+n\sum
h=0
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \sigma
h\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda + h,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
(2.13)
provided the other involved parameters are so constrained that each member can exist. Here
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given as in (2.12).
Theorem 9. Let z, \mu , \nu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu )\} > 0 and | z| < 1. Also let \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\sigma ), \Re (\xi ),
\Re (\zeta )\} > 0 and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1. Further, let x \in \BbbR +, C \in \BbbR +, and
\Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in (1.1). Then the
following formula holds true:
x\int
0
(t)\nu - 1(x - t)\mu - 1L(a\prime ,b\prime )
m (\zeta ;\sigma (x - t))L(c\prime ,d\prime )
n (\xi ;\sigma (x - t))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(x - t) - C
\bigr]
dt = x\mu +\nu - 1
m+n\sum
h=0
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime \sigma
h xh\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\mu ,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
(2.14)
provided the other involved parameters are so constrained that each member can exist. Here
\nabla n,m, \xi , \zeta
a\prime ,b\prime ,c\prime ,d\prime is given as in (2.12).
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CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1167
3. Special cases. It is noted that the results in Section 2 are general enough to be specialized to
yield a large number of simpler integral formulas. Here we choose to present the following formulas.
Corollary 1. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration
path Li\gamma \infty in (1.1). Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
zu - C
\bigr]
du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
\times
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda ,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
.
provided the other involved parameters are so constrained that each member can exist.
Proof. Setting \zeta = \xi = 1 in (2.11), after a little simplification, we get the desired result.
Corollary 2. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let x, t \in \BbbR
with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1,
C \in \BbbR +, and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in
(1.1). Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda + h,C), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist.
Proof. Setting a\prime = c\prime = \xi = \zeta = 1 in (2.13) and using (1.5) to consider L1,b
n (1;x) =
= Z
(1,b)
n (x; 1), after a little simplification, we get the desired result.
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1168 P. AGARWAL, M. CHAND, J. CHOI
Corollary 3. Let z, \mu , \nu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu ), \Re (\sigma )\} > 0 and | z| < 1. Also let x \in \BbbR +
and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ), \Re (d\prime )\} > - 1. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen
number from the integration path Li\gamma \infty in (1.1). Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]n \aleph \alpha ,\beta
\rho k,\sigma k,\delta k;r
\bigl[
z(x - t) - C
\bigr]
dt =
= x\mu +\nu - 1
n\sum
h=0
( - n)h \sigma
h xh\times
\times \aleph \alpha ,\beta +1
\rho k+1,\sigma k+1,\delta k;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , p), (aj , Aj)1,\beta , [\delta j(ajk , Ajk)]\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\mu , p), [\delta j(bjk , Bjk)]\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist.
Proof. Setting a\prime = c\prime = 0 and \xi = \zeta = 1 in (2.14), and using some suitable identities in
Section 1 including (1.7) – (1.10), after a little simplification, we get the desired result.
When \delta 1 = . . . = \delta r = 1 in (1.1), the definition of the I -function is recovered (see [7]):
I[z] = Im,n
pk,qk;r
[z] =
= \aleph m,n
pk,qk,1;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (aj , Aj)1,n, [1(ajk , Ajk)]n+1,pk;r
=
(bj , Bj)1,m, [1(bjk , Bjk)]m+1,qk;r
\Biggr]
=
=
1
2\pi i
\int
L
\Omega m,n
pk,qk,1;r
(s)z - s ds, (3.1)
where z \in \BbbC \setminus \{ 0\} , i =
\surd
- 1 and \Omega m,n
pk,qk,1;r
(s) is defined in (1.2), and the integration path L can
be used as in (1.1). Otherwise, a new integration path for this (3.1) can be chosen. The existence
conditions for the integral (3.1) can be easily deduced from those of the \aleph -function (1.1) with
\delta 1 = . . . = \delta r = 1.
Then the integral formulas in Corollaries 1 – 3 can reduce to yield the following integral formulas
involving the I -function given in Corollaries 4 – 6, respectively.
Corollary 4. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration
path Li\gamma \infty in (1.1). Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times I\alpha ,\beta \rho k,\sigma k;r
\bigl[
zu - C
\bigr]
du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
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CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1169
\times
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times I\alpha ,\beta +1
\rho k+1,\sigma k+1;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , (ajk , Ajk)\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda ,C), (bjk , Bjk)\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist.
Corollary 5. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let x, t \in \BbbR
with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1,
C \in \BbbR +, and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in
(1.1). Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times I\alpha ,\beta \rho k,\sigma k;r
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
\times I\alpha ,\beta +1
\rho k+1,\sigma k+1;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\beta , (ajk , Ajk)\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\lambda + h,C), (bjk , Bjk)\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist.
Corollary 6. Let z, \mu , \nu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu ), \Re (\sigma )\} > 0 and | z| < 1. Also let x \in \BbbR +
and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ), \Re (d\prime )\} > - 1. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen
number from the integration path Li\gamma \infty in (1.1). Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]n I\alpha ,\beta \rho k,\sigma k;r
\bigl[
z(x - t) - C
\bigr]
dt =
= x\mu +\nu - 1
n\sum
h=0
( - n)h \sigma
h xh\times
\times I\alpha ,\beta +1
\rho k+1,\sigma k+1;r
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , C), (aj , Aj)1,\beta , (ajk , Ajk)\beta +1,\rho k;r
(bj , Bj)1,\alpha , (\mu ,C), (bjk , Bjk)\alpha +1,\sigma k;r
\Biggr]
provided the other involved parameters are so constrained that each member can exist.
Further, the special case r = 1 of the I -function (3.1) reduces to become the H -function (see [3,
4]). Then the formulas in Corollaries 4 – 6 reduce to yield the following integral formulas involving
the H -function which are in Corollaries 7 – 9, respectively.
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1170 P. AGARWAL, M. CHAND, J. CHOI
Corollary 7. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration
path Li\gamma \infty in (1.1). Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times H\alpha ,\beta
\rho 1,\sigma 1
\bigl[
zu - C
\bigr]
du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
\times
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times H\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\lambda ,C)
\Biggr]
(3.2)
provided the other involved parameters are so constrained that each member can exist.
Corollary 8. Let z, \delta , \lambda \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let x, t \in \BbbR
with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1,
C \in \BbbR +, and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen number from the integration path Li\gamma \infty in
(1.1). Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times H\alpha ,\beta
\rho 1,\sigma 1
\bigl[
z(u - t) - C
\bigr]
du = \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
\times H\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h,C), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\lambda + h,C)
\Biggr]
(3.3)
provided the other involved parameters are so constrained that each member can exist.
Corollary 9. Let z, \mu , \nu \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu ), \Re (\sigma )\} > 0 and | z| < 1. Also let x \in \BbbR +
and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ), \Re (d\prime )\} > - 1. Further, let C \in \BbbR + and \Re (\lambda ) > - C \gamma where \gamma \in \BbbR is the chosen
number from the integration path Li\gamma \infty in (1.1). Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]nH\alpha ,\beta
\rho 1,\sigma 1
\bigl[
z(x - t) - C
\bigr]
dt =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1171
= x\mu +\nu - 1
n\sum
h=0
( - n)h \sigma
h xhH\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu , C), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\mu ,C)
\Biggr]
(3.4)
provided the other involved parameters are so constrained that each member can exist.
It is noted that the special case of the H -function when Aj = 1, j = 1, . . . , p, and Bj = 1,
j = 1, . . . , q, reduces to the Meijer’s G-function (see, e.g., [2], Section 8.2) as follows:
H\alpha ,\beta
\rho 1,\sigma 1
\biggl[
x
\bigm| \bigm| \bigm| \bigm| (aj , 1)1,\rho 1(bj , 1)1,\sigma 1
\biggr]
= G\alpha ,\beta
\rho 1,\sigma 1
\biggl[
x
\bigm| \bigm| \bigm| \bigm| (a\rho 1)(b\sigma 1)
\biggr]
.
Then the formulas in Corollaries 7 – 9 are seen to reduce to give the corresponding integral formulas
involving the Meijer’s G-function (3.5) – (3.7).
Corollary 10. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times G\alpha ,\beta
\rho 1,\sigma 1
[zu - 1] du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
\times
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times \Gamma (\lambda + \delta + h)
\Gamma (\lambda )
G\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h), (a\rho 1)
(b\sigma 1), (\lambda )
\Biggr]
(3.5)
provided the other involved parameters are so constrained that each member can exist.
Corollary 11. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
x, t \in \BbbR with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times G\alpha ,\beta
\rho 1,\sigma 1
[z(u - t) - 1] du = \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1172 P. AGARWAL, M. CHAND, J. CHOI
\times \Gamma (\lambda + \delta + h)
\Gamma (\lambda + h)
G\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\lambda + \delta + h), (a\rho 1)
(b\sigma 1), (\lambda + h)
\Biggr]
(3.6)
provided the other involved parameters are so constrained that each member can exist.
Corollary 12. Let z, \mu , \nu , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu ), \Re (\sigma )\} > 0 and | z| < 1. Also let
x \in \BbbR + and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ), \Re (d\prime )\} > - 1. Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]nG\alpha ,\beta
\rho 1,\sigma 1
[z(x - t) - 1] dt =
= x\mu +\nu - 1
n\sum
h=0
( - n)h \sigma
h xhG\alpha ,\beta +1
\rho 1+1,\sigma 1+1
\Biggl[
z
\bigm| \bigm| \bigm| \bigm| \bigm| (\mu + \nu ), (a\rho 1)
(b\sigma 1), (\mu )
\Biggr]
(3.7)
provided the other involved parameters are so constrained that each member can exist.
Here, replacing \sigma 1, aj , bj by \sigma 1+1, 1 - aj , 1 - bj with b1 = 0, respectively, and letting \alpha = 1
in the H -function is seen to yield the Wright’s generalized hypergeometric function p\Psi q (see, e.g.,
[14, p. 50]):
H1,\rho 1
\rho 1,\sigma 1+1
\biggl[
- x
\bigm| \bigm| \bigm| \bigm| (1 - aj , Aj)1,p
(0, 1), (1 - bj , Bj)1,q
\biggr]
= \rho 1\Psi \sigma 1
\biggl[
(aj , Aj)1,p ;
(bj , Bj)1,q ;
x
\biggr]
. (3.8)
Then, applying the relation (3.8) to the formulas (3.2), (3.3) and (3.4) yields the following correspon-
ding integral formulas involving the Wright’s generalized hypergeometric function p\Psi q (3.9) – (3.11).
Corollary 13. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times \rho 1\Psi \sigma 1 [zu
- p] du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
\times
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times \rho 1+1\Psi \sigma 1+1
\Biggl[
(\lambda + \delta + h, p), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\lambda , p)
; z
\Biggr]
(3.9)
provided the other involved parameters are so constrained that each member can exist.
Corollary 14. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
x, t \in \BbbR with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )\} > - 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1173
Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times \rho 1\Psi \sigma 1 [z(u - t) - p] du = \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
\times \rho 1+1\Psi \sigma 1+1
\Biggl[
(\lambda + \delta + h, p), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\lambda + h, p)
; z
\Biggr]
(3.10)
provided the other involved parameters are so constrained that each member can exist.
Corollary 15. Let z, \mu , \nu , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ), \Re (\nu ), \Re (\sigma )\} > 0 and | z| < 1. Also let
x \in \BbbR + and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ), \Re (d\prime )\} > - 1. Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]n \rho 1\Psi \sigma 1 [z(x - t) - p] dt =
= x\mu +\nu - 1
n\sum
h=0
( - n)h \sigma
h xh\rho 1+1\Psi \sigma 1+1
\Biggl[
(\mu + \nu , p), (aj , Aj)1,\rho 1
(bj , Bj)1,\sigma 1 , (\mu , p)
; z
\Biggr]
(3.11)
provided the other involved parameters are so constrained that each member can exist.
Also, choosing p = 1; \alpha = 1, \beta = 2, \rho 1 = \sigma 1 = 2; Aj = Bj = 1; b1 = 0 and replace a1, a2,
b2 into 1 - a1, 1 - a2, 1 - b2, respectively, the H -function reduces to the Gaussian hypergeometric
function 2F1 as follows:
H1,2
2,2
\biggl[
x
\bigm| \bigm| \bigm| \bigm| (1 - a1, 1), (1 - a2, 1)
(0, 1), (1 - b2, 1)
\biggr]
=
\Gamma (a1)\Gamma (a2)
\Gamma (b2)
2F1[a1, a2; b2; - x]. (3.12)
Then applying the relation (3.12) to the formulas (3.2), (3.3) and (3.4) is seen to yield the follo-
wing results (3.13) – (3.15) whose integrands and resulting formulas contain 2F1 and the generalized
hypergeometric function 3F2, respectively.
Corollary 16. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ), \Re (\lambda ), \Re (\sigma )\} > 0 and | z| < 1. Also let
\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )
\bigr\}
> - 1.
Then the following formula holds true:
1\int
0
u\lambda - 1(1 - u)\delta - 1L(a\prime ,b\prime )
m (\sigma (1 - u))L(c\prime ,d\prime )
n (\sigma (1 - u))\times
\times 2F1[a1, a2; b2; zu
- 1] du =
\Gamma (a\prime n+ b\prime + 1)\Gamma (c\prime m+ d\prime + 1)
m!n!
m+n\sum
h=0
\sigma h \Gamma (\delta + h)\times
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1174 P. AGARWAL, M. CHAND, J. CHOI
\times (\delta + h)\lambda
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)h - k( - n)k
\Gamma (a\prime k + b\prime + 1)\Gamma (c\prime (h - k) + d\prime + 1)
\biggr]
\times
\times 3F2[\lambda + \delta + h, a1, a2; b2, \lambda ; z] (3.13)
provided the other involved parameters are so constrained that each member can exist.
Corollary 17. Let z, \delta , \lambda , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\delta ),\Re (\lambda ),\Re (\sigma )\} > 0 and | z| < 1. Also let x,
t \in \BbbR with x \geq t. Further, let
\mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\Re (a\prime ), \Re (b\prime ), \Re (c\prime ), \Re (d\prime )
\bigr\}
> - 1.
Then the following formula holds true:
x\int
t
(x - u)\delta - 1(u - t)\lambda - 1Z(1,b\prime )
m (\sigma (u - t); 1)Z(1,d\prime )
n (\sigma (u - t); 1)\times
\times 2F1[a1, a2; b2; z(u - t) - 1] du =
= \Gamma (\delta )(x - t)\delta +\lambda - 1\Gamma (n+ b\prime + 1)\Gamma (m+ d\prime + 1)
m!n!
\times
\times
m+n\sum
h=0
\sigma h (\lambda + h)\delta
h\sum
k=0
\biggl(
h
k
\biggr) \biggl[
( - m)(h - k)( - n)k
\Gamma (k + b\prime + 1)\Gamma ((h - k) + d\prime + 1)
\biggr]
\times
\times 3F2[\lambda + \delta + h, a1, a2; b2, \lambda + h; z] (3.14)
provided the other involved parameters are so constrained that each member can exist.
Corollary 18. Let z, \mu , \nu , \sigma \in \BbbC with \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (\mu ),\Re (\nu ),\Re (\sigma )\} > 0 and | z| < 1. Also let
x \in \BbbR + and \mathrm{m}\mathrm{i}\mathrm{n}\{ \Re (b\prime ),\Re (d\prime )\} > - 1. Then the following formula holds true:
x\int
0
t\nu - 1(x - t)\mu - 1 [1 - \sigma (x - t)]n 2F1[a1, a2; b2; z(x - t) - 1] dt =
= x\mu +\nu - 1(\mu )\nu
n\sum
h=0
( - n)h \sigma
h xh3F2[\mu + \nu , a1, a2; b2, \mu ; z] (3.15)
provided the other involved parameters are so constrained that each member can exist.
References
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Halsted Press (John Wiley \& Sons), 1978.
4. Mathai A. M., Saxena R. K., Haubold H. J. The H -function: theory and applications. – New York: Springer, 2010.
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CERTAIN INTEGRALS INVOLVING \aleph -FUNCTIONS AND LAGUERRE POLYNOMIALS 1175
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Received 12.08.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
|
| id | umjimathkievua-article-1507 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:10Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ee/d0a547f5971370e7c0fe9920600bcfee.pdf |
| spelling | umjimathkievua-article-15072019-12-05T08:57:49Z Certain integrals involving ℵ-functions and Laguerre polynomials Деякi iнтеграли, що включають ℵ -функцiї та полiноми Лагерра Agarwal, P. Chand, M. Choi, J. Агарвал, Р. П. Чанд, М. Чой, Дж. UDC 517.5 Our aim is to establish certain new integral formulas involving $\aleph$ -functions associated with Laguerre-type polynomials. We also show how the main results presented in paper are general by demonstrating 18 integral formulas that involve simpler known functions, e.g., the generalized hypergeometric function $_pF_q$ in a fairly systematic way. УДК 517.5 Наша мета — встановити деякi новi iнтегральнi формули, що включають $\aleph$ -функцiї, асоцiйованi з полiномами лагеррiвського типу. Також показано, що основнi результати, отриманi у статтi, є загальними. Для цього наведено 18 iнтегральних формул, що включають бiльш простi вiдомi функцiї, наприклад узагальнену гiпергеометричну функцiю $_pF_q$ в досить загальному виглядi. Institute of Mathematics, NAS of Ukraine 2019-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1507 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 9 (2019); 1159-1175 Український математичний журнал; Том 71 № 9 (2019); 1159-1175 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1507/491 Copyright (c) 2019 Agarwal P.; Chand M.; Choi J. |
| spellingShingle | Agarwal, P. Chand, M. Choi, J. Агарвал, Р. П. Чанд, М. Чой, Дж. Certain integrals involving ℵ-functions and Laguerre polynomials |
| title | Certain integrals involving ℵ-functions and Laguerre
polynomials |
| title_alt | Деякi iнтеграли, що включають ℵ -функцiї
та полiноми Лагерра |
| title_full | Certain integrals involving ℵ-functions and Laguerre
polynomials |
| title_fullStr | Certain integrals involving ℵ-functions and Laguerre
polynomials |
| title_full_unstemmed | Certain integrals involving ℵ-functions and Laguerre
polynomials |
| title_short | Certain integrals involving ℵ-functions and Laguerre
polynomials |
| title_sort | certain integrals involving ℵ-functions and laguerre
polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1507 |
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