CERTAIN INTEGRALS INVOLVING GENERALIZED MITTAG-LEFFLER TYPE FUNCTIONS Текст научной статьи по специальности «Математика»

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CERTAIN INTEGRALS INVOLVING GENERALIZED MITTAG-LEFFLER TYPE FUNCTIONS

Sirazul Haqa, Maggie Aphaneb, Mohammad Saeed Khanc, Nicola Fabianod

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a J.S.University, Department of Applied Science,

Shikohabad, Firozabad, U.P., Republic of India, .g

e-mail: sirajulhaq007@gmail.com,

ORCID iD: ©https://orcid.org/0000-0001-9297-2445 b n

b Sefako Makgatho Health Sciences University, 10

Department of Mathematics and Applied Mathematics, ^

Ga-Rankuwa, Republic of South Africa,

e-mail: maggie.aphane@smu.ac.za

ORCID iD: ©https://orcid.org/0000-0002-9640-7846

o

c Sefako Makgatho Health Sciences University, >

Department of Mathematics and Applied Mathematics, Ga-Rankuwa, Republic of South Africa, e-mail: drsaeed9@gmail.com, g

ORCID iD: ©https://orcid.org/0000-0003-0216-241X

d University of Belgrade, "Vinca" Institute of Nuclear Sciences -Institute of National Importance for the Republic of Serbia, Belgrade, Republic of Serbia, e-mail: nicola.fabiano@gmail.com, corresponding author, ®

ORCID iD: ©https://orcid.org/0000-0003-1645-2071

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DOI: 10.5937/vojtehg70-40296;https://doi.org/10.5937/vojtehg70-40296 £

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FIELD: Mathematics

ARTICLE TYPE: Original scientific paper Abstract:

Introduction/purpose: Certain integrals involving the generalized Mittag-Leffler function with different types of polynomials are established.

Methods: The properties of the generalized Mittag-Leffler function are used in conjunction with different kinds of polynomials such as Jacobi, Legendre, and Hermite in order to evaluate their integrals.

Results: Some integral formulae involving the Legendre function, the Bessel Maitland function and the generalized hypergeometric functions are derived.

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Conclusions: The results obtained here are general in nature and could be useful to establish further integral formulae involving other kinds of polynomials.

Key words: Mittag-Leffler function, Generalized hypergeometric function, Bessel Maitland function, Jacobi polynomials, Hermite polynomials.

This paper follows the lines of the companion paper (Haq et al, 2019)

Introduction

This paper follo

g involving the generalized Galue-type Struve function in which the same ^ topics are dealt here with the generalized Mittag-Leffler functions. As it is ^ well known, a special function:

o

LU

~ ( )k

(z^EfTTk+n' z^ e C K(v) > 0, (1)

k=0 T(vk + 1)'

and its general form

(z) = J2T(\ + )' z,v £ C, ft(v) > 0, > 0, (2)

to k=0 (V + W)

<

are called Mittag-Leffler functions (Erdelyi et al, 1953a), C being the set of g complex numbers. The former was established by Mittag-Leffler (Mittag-Leffler, 1903) in connection with his method of summation of some diver-lju gent series. Certain properties of this function were studied and investi-o gated. The function defined by (2) appeared for the first time in the work of o Wiman (Wiman, 1905). The functions given by equations (1) and (2) are entire functions of order f = 1 and of type a = 1 (see for example (Erdelyi et al, 1953b)). By means of the series representations, a generalization of the functions defined by equations (1) and (2) is introduced by Prabhakar

(Prabhakar, 1971) as:

~ (P), zk

(z) = k=0 T(vk + u)k\, ^ C, v) > 0, > 0, (3)

where

(P)k = P(P +1) ••• (P + k - 1) = r(p(+k),

whenever r(P) is defined, (p)0 = 1,p = 0. It is an entire function of order f = (1/v)[^(v)R(v^]-1/v. For various properties of this function with applications, see Prabhakar (Prabhakar, 1971). Further generalization of the

Mittag-Leffler function EP^(z) was considered earlier by Shukla and Pra-japati (Shukla & Prajapati, 2007) which is given as:

fP,q (z) = V (P)kqZk

k= T(vk + w)kV with z,u,p e C, K(u) > max(0, ft(q) - 1), ft(q) > 0, (4)

= t • (5)

P*r\v) = 2*1

-n, 1 + q + a + n; 1 - y q +1; 2

(6)

When q = a = 0, the polynomial in (6) becomes the Legendre polynomial (Rainville, 1960). From (6), it follows that Pn'a)(y) is a polynomial of

oo I

en

£± CP

<u

CP

which is the special case when q e (0,1) and min{K(w), K(p)} > 0. ^

In continuation of this study, Salim and Faraj (Salim & Faraj, 2012; Nadir et al, 2014) introduced a new generalization of the Mittag-Leffler function which was given as:

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N "rö

k=0 v /v " & (min{ft(u), &(p), K(5)} > 0; p,q> 0; z,v,u,p,S, e C).

Numerous generalizations and cases of the Mittag-Leffler function have o

been studied and investigated, see for details (Singh & Rawat, 2013; Wright, 1935b; Faraj et al, 2013; Dorrego & Cerutti, 2012; Srivastava & Tomovski, 2009; Saxena etal, 2011; Khan & Ahmed, 2012).

Integral formulae involving the Mittag-Leffler functions have been developed by many authors, see for example, (Prajapati & Shukla, 2012; Prajapati et al, 2013; Gehlot, 2021; Purohit et al, 2011). In this sequel, here, we aim to establish certain new generalized integral formulae involving the new generalization of the Mittag-Leffler function. The main result presented here is general enough to be specialized to give many interesting integral formulae which are derived as special cases.

Integrals with the Jacobi polynomials

The Jacobi polynomialsP,f'CT(y) (Rainville, 1960; Srivastava & Manocha, 1984) may be defined by

ro <u OT IT

ro X

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CN CM o CM

, (7)

n !

the degree n and that

P^(V) = ^^ -Here, we obtain the following integrals.

Theorem 1. If p,q > 0 z,v,u,p,S, £ C, %(v) > 0, > 0, %(p) > 0, %(5) > 0 and % formula holds true

E 0 , > 0 and №(() > —1, g > —1,a > —1 then the following integral

0£

o

0

<5 J 1 Vi (1 — y)e(1 + y)v+hkPne'a)(y)ES:t]qP[z(1 + y)h]dy

1

o

LU

(—1)n2e+n+lr(n + hk + 1)r(n + g + 1)r(n + hk + a + 1)

>-

xi,

ft X sp:t:qp(2hz) x 3F2

n!r(n + hk + a + n + 1)r(n + hk + g + n + 2) + hk + a + 1,n + hk + 1;

n + hk + a + n + 1,n + hk + g + n + 2; 1

(8)

Proof. Naming the left-hand side (LHS) of (8) as h and using the defi-

co

u nition (5), we have

h = J1 rf (1 — y)e(1 + y)n Pne'a)(y)ES:t]qp[z(1 + y)h]dy 0 '1 — (P)kq [Z(1+ y)h]k

/1 — _ 1y—ym+yrnz r(uk+„mPt

dy,

interchanging the order of integration and summation which is permissible under the conditions of the theorem, the above expression becomes

g riï^u - v —y)e{1+y)V+hk Pr'my■

Apply the following formula (Saxena, 2008) on (9)

r 1

(9)

yi (1 — yY(1 + y)n Pne'a)(y)dy =

'-1

(—1)n2e+v+1r(n + 1)r(n + g + 1)r(n + a + 1) n!r(n + a + n + 1)r(n + g + n + 2)

800

X3F2

n + a + 1, n + 1;

1

n + a + n + 1, n + Q + n + 2; provided that q> -1 and a > -1, and we get the desired result.

(10)

Theorem 2. If p,q > 0 z,v,u,p,S, e C, ft(u) > 0, > 0, ft(p) > 0, K(^) > 0 and > —1, q > -1, a > -1 then the following integral formula holds true

f (1 - y)n(1 + y)°Pne'a)(y)Pn^)(y)EP:t]qp[z(1 - y)h]dy

= 2n+g+1r(1 + v + m)r(1 + q + n) mini

(-m)k(1 + v + d + m)k ^ (-m)i(1 + v + d + m)i

X k= r(1+ v + k)(ki) ¿0 r(1 + v + k)(ll)

xEP'^'q,(2hz)B(1 + n + hk + k + l, 1 + a).

(11)

Proof. Denoting the LHS of (11) by I2 and using definition (5), we get

12 = J V - y)n (1 + y)° Pne'a)(y)Pn^)(y)E£:tjqp[z(1 - y)h]dy

12 = E

(p)kq(z)k f1 = T(vk + u)(S)pk J-i

(1 - y)v+hk(1 + yyPnS'-)(y)Pn^i'e)(y)dy. (12)

Now, using (6) in (12), we get

" (p) kq (z)k (1+ H)r

I2 = E

E

(-m)k (1 + v + d + m)k

k=o r(uk + u)(5)Pk mi k=0

(1+ ¡j)k,2k kl X J 1(1 - y)n+hk+k(1 + yf Pne'a)(y)dy. Again using (6) in (13), we attain

" (p) kq (z)k r(1+ v + m)r(1 + Q + n)

(13)

12 = E

k=0

r(uk + u)(5)pk

m!n!

(-m)k(1 + v + d + m)k ^ (-m)i(1 + v + d + m)i k=or(1 + v + k)2k (kl) ¿0 r(1 + v + l)2l (ll)

00 1

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7.

p. p

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flf e

L -

g

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n

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er

n

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lv ol

v in

ls

ra gr

e

e C

a t e

oo

IT

a

X

801

^r n 1

- x J (1 - y)n+hk+k+l(1 + vTPne'a)(v)dy, (14)

o

o >

< O

CD

S2 ■O

x

LU I—

o

o >

-1

but by the formula found in (Rainville, 1960; Srivastava & Manocha, 1984)

J1 (1 - y)v+n(1 + y)a+ndy = 22n+a+n+1B(1 + n + n, 1 + a + n), (15)

R J-i

yy using it in (14), we get the required result.

CC

o o

Theorem 3. If p,q > 0 z,v,u,p,5, £ C, %(v) > 0, > 0, %(p) > 0, %(5) > 0 and q> -1, a > -1, then

1

o I (1 - yY(1 + yf Pne,a)(y)ES:t,qP[z(1 - y)h(1 + y))]dy

w J-1

>- =2^+1(±+o)n A (-n)k (1 + Q + a + n)k

* n! n! k=0 (1 + Q)k (k!)

x£p'sjqp(2h+)z)M(1 + f + hk + k, 1 + d + tk). (16) Proof. Denoting the LHS of (16) by I3,

13 = J V - yr(1 + y) Pne: a)(y)£pvtl [z(1 - y)h(1 + y))]dy

£ rrak L1(1 - y)M+hk (1+y)e+)k a)(y)dy

(17)

k=0

Now, using (6) in (17) we have

(P)kq(z)k (1 + Q)n A (-n)k(1 + Q + a + n)k

I \ " (P)kq(z) (1 + Q)n y^

k=0 T(vk + u)(5)pk n! k=0 (1 + Q)k2kk!

x J1 (1 - y)n+V+hk+k-n(1 + y)n+e+)k-ndy,

further, using (15) in (18) we attain the desired result.

(18)

Theorem 4. If p,q > 0 z,v,u,p,5, £ C, %(v) > 0, K(w) > 0, %(p) > 0, %(5) > 0 and q > -1, a > -1, then

f (1 - vT(1 + v)P{ne:a)(y)ES:!]qp[z(1 + y)-h]dy

2^+e+1 (1 + Q)n ~ (-n)k(1 + q + a + n)k

nl

nl

E

k=o

(1 + Q)k (kl)

xEp%(2-hz)B(1 + v + k, 1 + 9 - hk).

(19)

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7 9

7.

p. p

Proof. Denoting the LHS of (19) by I4,

I4 = /1(1 - y)M(1 + y)*Pne'a)(y)E£:t]qp[z(1 + y)-h]dy

-1

<x

E

k=o

(p)kq(z)k

r(uk + u)(5)pk J-i

(1 - y)^(1+ y)*-hkPne'a)(y)dy.

Now, using (6) in (20) we attain

" (p) kq (z)k (1 + Q)

E

=0 r(uk + u)(5)pk nl

k=o

E

k=o

(-n)k (1 + Q + a + n)k (1 + Q)k 2k kl

xj (1 - y)n+M+k-n(1 + y)n+*

-hk-n

dy

further, using (15) in (21) we attain the required result.

(20)

(21)

Theorem 5. If p,q > 0 z,v,u,p,S, e C, ft(u) > 0, > 0, ft(p) > 0, > 0 and q > -1, a > -1, then

(1 - y)"(1 + y)*P^rt(y)EP:t]qp[z(1 - y)h(1+ y)-t]dy

1

oo

2M+0+1 (1 + Q)n ^ (-n)k(1 + Q + a + n)k

nl

nl

k=o

(1 + Q)k (kl)

xEp'sjqp(2h-tz)B(1 + v + hk + k, 1 + 9 - tk).

(22)

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ler

flf e

L -

g

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d

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n

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lv ol

v in

ls

ra gr

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a t e

OT IT

a

X

Proof. Denoting the LHS of (22) by I5,

I5 = / i(1 - y)M(1 + y)* Pne'a)(y)EP:t]qp[z(1 - y)h(1 + y)-t]dy

E

k=o

(p)kq(z)k

r(uk + u)(5)pk J-i

(1 - yT+hk(1 + y)*-tkP^rt(y)dy,

(23)

1

4

1

1

<D

now using (6) in (23) we attain

CD

S2 >o

X LU I—

o o

J \ - (p)kq(z)k (1 + Q)n V^ (-n)fc(1 + Q + + n)k

_ k= T(vk + u)(5)pk n! (1 + Q)k2kk!

x j (1 - y)n+M+^k+k-n(i + y)n+°-tk-ndy (24) further, using (15) in (24) we attain the required result.

Some special cases

If we replace n by £ - 1 and put q = a = n = d = 0 then the integral nsforms into (Rainville, 1960)

J6 = J 1(1 - y?-1Pn(y)S$qp[z(1 - y)^]dy

^ (-m)k(1 + m)k ^ ( n)i(1 + n)i

16 = ^-^-x.L

o I2 transforms into the following integral involving the Legendre polynomial

i>-

CC <

CO

^ — (k!)2 ^ l!2

5 k=0 v ' l=0

x £p:t]qp(2h z)B(1 + £ + hk + k + l, 1). (25)

If a = q = 0, n is replaced by n - 1 and d by d - 1, then the integral I3 transforms into the following integral involving the Legendre polynomial (Rainville, 1960)

& it = j 1(1 - yr-1(1 + y)0-1Pn(y)£P:t;qp[z(1 - y)h(1 + y)t]dy

2^+^-i(-n)k (1 + n)k

It = E

(k!)

2

k=0

x£p'Sjqp(2h+tz)B(1 + n + hk + k, 0 + ik). (26)

If q = a = 0, n is replaced by n - 1 and d by d - 1 then the integral I3 transforms into the following integral involving the Legendre polynomial (Rainville, 1960)

Is = J 1(1 - yr-1(1 + y)d-1Pn(y)££:t]qp[z(1 - y)h(1 + y)-t]dy

I8 =

E

k=0

2^d-l(-n)k (1 + n)k (k!)2

x£^qp(2h-tz)B(1 + f + hk + k,e - tk).

(27)

oo I

en

S± CP

Integral with the Bessel Maitland function

The special case of the Wright function (Erdelyi et al, 1953b), see also (Wright, 1935a,b) written in the form

(-; A, a; z) = o^i

(A, a);

E

k=0

1

r(Ak + a) kV

(28)

with complex z,a e C and real A e R. When A = n,a = v + 1 and z is replaced by -z, then the function v + 1; -z) is defined by JV(z)

t(rj,v + 1; -z) = jn(z) = J2

1

(-z)k

k=0

r(nk + v + 1) k! '

(29)

and such a function is known as the Bessel Maitland function, or the generalized Bessel function, or the Wright generalized Bessel function, see (Mcbride, 1995).

Theorem 6. If p,q > 0 z,v,u,p,S, e C ,&(v) > 0, №(u) > 0, №(p) > 0, №(5) > 0, q - qt > -1, q > 0, 0 < t < 1 and + 1) > 0, then the following integral formula holds true.

- (vYJl(v)£^l[z(yy]dy = r1 +V-.+ 'iek)) x £P^l(z)- (30)

i

Proof. Naming the LHS of (30) as I9, we obtain

<u p

It <u

-J

I

OT

ro

"O

<u n

lo

<D

c

<u

_>

o

>

C

m

<o <u

<u O

ro <u OT IT

ro X

I9 = / (vYJl(v)&z(v)e]dy

1-1

'9 = E

(P)qk z

k

X (y^k)JT (v)dy. 0

k=0 r(uk + W)(5)pk Jo Now we know the formula, see (Saxena, 2008)

Jo °° (vJ (v)dv = m+f+r- f

(31)

(32)

k

z

z

<D

provided K(p) > -1,0 <t < 1.

o

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cn

CM o CM

0£ yy

0£ ZD

o

O

-J

<

o

x

o

LU

I—

>-

CC <

Using (32) in (31), we attain

(P)qk Zk

I9 = E

r(p + gk + 1)

k=0

r(vk + u)(S)pk r(1 + v - t - t (p + gk))''

hence proved.

Integrals with the Legendre functions

The Legendre functions are solution of Legendre's differential equation, see (Erdelyi et al, 1953a)

(1 - z2) - + [(v(v + 1)) - u2(1 - z2)-1]w = 0, (33) dz2 dz

where z, v, u are unrestricted.

CD

S2 ■O

x

LU I—

o

o

Under the subsitution w = (z2 - 1)w/2v in (5.1) becomes

d2

dv

(1 - z2)db? - 2(u + 1)zdz + [(v + u)(v + u + 1)]v = 0,

dz

(34)

and with A = 1/2-z/2 as the independent variable, this differential equation becomes

d2 v dv

A(1 - A)dA2 + (u + 1)(1 - 2A)dA + [(v - u)(v + u + 1)]v = 0. (35)

This is the Gauss hypergeometric type equation with a = u - v,b =

v + u + 1,c = u + 1.

Hence it follows that the function

W = (z) =

1

fz±1)

r(1 - u)\z - 1J

w/2

2F1

-v, v + 1; 1 - u;

1/2 - z/2

(36)

for I 1 - z \< 2 is a solution of (33).

The function (z) is known as the Legendre function of the first kind (Erdelyi et al, 1953a). It is one valued and regular on the z-plane, supposed cut along the real axis from 1 to -to.

806

Theorem 7. If p,q > 0 z,v,u,p,5, e C, №(u) > 0, №(u) > 0, №(p) > 0, №(5) > 0 and 0 > 0 and n is a positive integer then

oo I

en

8—1/1 n,2\v/2r>Wf„,\eP,S,q \pijni Q!

(v)e-l(1 - v2)n/2Pn(y)&z(v)e]dy (-1)n r(v + n + 1)

■ x

r(1 - n + v)

r(0 + Qk) CP , 5 , /07X &

v_r(0 + Qk)_fp,q (?/2o) (37) _

k= r{1/2 + ^ + n/2 - v/2)r(1 + ^ + n/2 + v/2) tv'^p(z' )' () |

_J i

Proof. Denoting the LHS of (37) by Iio, f

'io = f \y)e—1(1 - y2)n/2Pn(v)ES:t]qP[z(y)e]dy 0

'io = E r( P*zk5) x ^ ye—1+ek(1 - v2)n/2Pn(v)dy. (38) & t^r(vk + ^)(5)pk Jo

(39)

Now (38) becomes

" (P)qkzk

k=0 T(vk + u)(5)pk

'io = E

_(-1)n ^n2-(e+ek—r(d + Qk)r(n + v + 1)_

r(1/2 + M + n/2 - v/2)r(1 + M + n/2 + v/2)r(1 - n + v), which is the desired result.

Theorem 8. If p,q > 0 z,v,u,p,5, e C, №(u) > 0, №(u) > 0, №(p) > 0, №(5) > 0 and 0 > 0 and n is a positive integer then

f (y)8—1(1 - y2)—n/2Pn(y)&z(v)e]dy

o

"O

<u .N

k=o

Now the integral in (38) can be solved by using the formula (Erdelyi et

al, 1953a)

/ ye—1(1 - y2)v/2Pn(v)dy f

o

_ (-1)n Vn2—d—n r(6)r(v + n + 1)

<u

(r(1/2 + 0/2 + n/2 - v/2)(r(1 + 0/2 + n/2 + v/2)(1 - n + v)' v ' o provided №(0) > 0,n = 1,2,3,....

ro <u 00 IT

ro X

i

X

o

"(5 >

cn

CM o CM

>-

CC <

O >

k=0 r(1/2 + ^ - v/2 - v/2)r(1 + M - n/2 - v/2)

xS^^). (40)

Proof. Denoting the LHS of (40) by In,

I11 = I\v)d-1(i - v2)-v/2PU(v)££:t]qP[z(v)e]dy Jo

1

Qi T I f„,\d-1fi „.2\-n

yy

CC J0

Ill = E TdkfqAm x i ve-1+ek(1 - v2)-v/2py(v)dy, (41)

< k=o r(uk + w)(^)pk Jo

i now the integral in (41) can be solved by using the formula (Erdelyi et al, lu 1953a)

roo

/ /-1(1 - v2)-v/2PU(v)dv

o

^n2-e+n T(d)

r(1/2 + e/2 - n/2 - v/2)r(1 + e/2 - n/2 - v/2)'

(42)

3 provided ft(0) > 0,n = 1,2,3,.... Again (41) becomes

i

LU ^o f \ k

q j V (P )qkzk

z 11 k=or(uk + w)(i)pk

(p)qkzk

k + w)(

r(1/2 + M - n/2 - v/2)r(1 + ^ - n/2 - v/2)'

Integrals with the Hermite polynomials

The Hermite polynomials Hn(v), see (Rainville, 1960; Srivastava & Manocha, 1984) may be defined by means of the relation

exp(2vt - t2) = £ ^^, (43)

k=0 '

valid for all finite v and t. Since

exp(2vt - t2) = exp(2vt) exp(-t2)

808

(2y)ntn^ (-1)kt

y^ (2v)

Z-^i n! Z-^i

k 2k

oo I

n=o k=o cB

^ \n/<2\ / 2kn £

n! ' k!

k=o

(-1)k (2y)n—2k t

^^ (n - 2k)!k!

n=o k=o

It follows from (43) that

<u

n2 (-1)k(2y)n—2ktn f

k=o (n - 2k)!k! ' () §

■

The examination of equation (44) shows that Hn(y) is a polynomial of | degree precisely n in y and that

"O

<u

Hn(y) = 2nvn + nn—2(y) (45) 1

<u c

in which nn—2(y) is a polynomial of the degree (n - 2) in y. &

Theorem 9. If p,q > 0 z,v,u,p,5, e C, №(v) > 0, №(w) > 0, №(p) > 0, №(5) > 0 and h > 0 №(p.) = 0,1,2 ... then

(y)2» exp(-y2)H2v(y)£^qp[z(y)—2h]dy

T(2V - hk + ^ w fP,5,q (22h z) - hk - v + 1) X Ev'"P(2 z)■

k=o

(46)

(47)

now the integral in (47) can be solved by using the formula (Saxena, 2008)

LV' ^ ^ = T(» - v+1) ■ (48)

Again (47) becomes

' = ^ (p)qkzk ^22(v—(2»+2hk))T(2y - 2hk + 1)

12 = k=o r(uk + u)(5)pk X T(» - hk - v + 1) .

_>

o

>

C V) 2

o

Proof. Denoting the LHS of (9) by I12, we have

/oo

(y)2» exp(-y2)H2v(y)SP:5]qp[z(y)—2h]dy *

-oo

'12 = E rlUk+qmPk xjZ ^H2V(y)1v-

v (y)Ev.u,p[z(y) ]dy

oo

<D

O

< O

>-

Q1 <

<5 Integrals with the generalized hypergeometric functions

CD

S2 A generalized hypergeometric function (Rainville, 1960) may be defined ° by

X LU I—

o

—)

o

Theorem 10. If p,q > 0 z,v,u,p,S, e C, > 0, &(w) > 0, K(p) > 0, > 0 and h > 0 ft(^) = 0,1,2... then

(y)2^ exp(-y2)H2v (y)£S:t]qP[z(y)2h]dy

O I \y J CAr\ y ) ±J-2V\y JEV:^:P

f g $+hkI, x^ (^z>.

s

o o

(49)

Proof. Denoting the LHS of (10) by '13, we have

Ii3 = (y)2M exp(-y2)H2v (y)£P:t]qp[z(y)2h]dy J — <x

O ^ (p)qk zk

'13 = grJ^ X L (y)dy, (50)

using the formula mentioned in (48), then the above expression (50), we get the desired result.

pFq

(Q)i, (Q)2,.....(Q)p; z

(a)1, (a)2,.........(a)q, ; .

y YlIj=i(6i)n zn (51)

¿0 nq=i(^i )n n!' (51)

in which no denominator parameter aj is allowed to be zero or a negative integer. If any numerator parameter ^ in (51) is zero or a negative integer, the series terminates.

Theorem 11. The following integral formula holds true,

f (yr-1(t - y)d-1PFq[(lp); (mq) : ay?(t - y)°]SP'^zyu(t - y)v]dy Jo

oo

= £vSJqP(ztu+v f (k)t(e+a)k x B(p + uk + 6k,9 + vk + ak), (52)

o,p\

k=0

oo

oo

where

f(k) = ( ) (53)

(mi)k,......(mq)"

f y^+uk-l(1 - x/t)d+vk-\Fq[(lp); (mq) : ay*(t - y)°]dy, J 0

/i5 = g (p)qk (ztu+v)"t^+e-1 x

[\sY+uk-l(l - s)e+vk-lpFq[(lp); (mq) : ats+ase(1 - s)CT]ds. 0

o,p\

k=0

(54)

where f (k) is defined in (53) provided

811

oo I

en

S± Cp

provided

(1) №(u) > 0, №(u) > 0, №(p) > 0, №(5) > 0 and p,q> 0,

(2)№(q) > 0, №(v) > 0 (both are not zero simultaneously),

(3) q and a are positive integers such that q + a > l. ^

JD ¡t=

Proof. Representing the LHS of (11) by Il5, we have 0

ro

rt £

)d-lpFq[(lp); (mq) : aye(t - yf ]£pv^p[zyu(, - y, ^y 'o |

V (P)qkz"te+Vk-l %

l k=0T(vk + U)(5)pk |

t

> o

'0 |

m

putting x = st and dx = tds, then we get 2

<u

k=o T(Vk + U)(5)pk s

o

(mq): atKI s"(i - s) ]ds. w

10 "<D

oo

The remaining theorems could be proved in a completely analogous fash- & ion. 1

Theorem 12. The following integral formula holds true,

Il6 = f(yY-l(t - y)e-lpFq[(lp); (mq) : ay*(t - y)°]£pv^zy-u(t - y)-v]dy 0

oo

Il6 = £^qp(zt-u-V)t^+e-lY, f (k)t(e+a)k x - uk + Qk,e - vk + ak),

<D

(1) > 0, > 0, K(p) > 0, &(5) > 0 and p,q> 0,

(2)№(q) > 0, №(v) > 0 (both are not zero simultaneously), o~ (3) q and a are positive integers such that q + a > 1.

>-

OH <

Theorem 13. The following integral formula holds true,

117 = f(vY-l(t - y)e-\Fq[(lp); (mq) : ay?(t - y)°\Epv%\zyu(t - y)-v]dy Jo

oo

117 = £^qv(ztu-v)t^+e-lY^ f (k)t(e+a)k x B(p + uk + Qk,e - vk + ak),

k=0

o (55)

o where f (k) is defined in (53) provided

(1) K(u) > 0, K(w) > 0, K(p) > 0, > 0 and p,q> 0,

(2)№(q) > 0, K(v) > 0 (both are not zero simultaneously),

(3) q and a are positive integers such that q + a > 1.

w Theorem 14. The following integral formula holds true,

o

lis = I (y)M-1(t - y)d-1PFq[(lp); (mq) : nyS(t - y)°]f.PM [zy-u '0

X LU I—

o

lis = / (yr-1(t - y)0-1pFq[(lp); (mq) : ay?(t - yf ^p[zy-u(t - y)v]dy Jo

oo

lis = £p:t]qp(zt-u+vf (k)t{e+a)k X B(p - uk + Qk,e + vk + ak),

k=o

O (56)

provided

(1) K(u) > 0, K(w) > 0, K(p) > 0, > 0 and p,q> 0,

(2)№(q) > 0, K(v) > 0 (both are not zero simultaneously),

(3) q and a are positive integers such that q + a > 1.

Conclusions

Certain new generalized integral formulae involving the Generalized Mittag-Leffler Type functions with many types of polynomials were established in this study. The results obtained here are general in nature and yield to many interesting formulae which are derived as particular cases.

References

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Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F.G. 1953b. Higher transcendental functions Volume 3 (Bateman Manuscript Project) [online]. New York, Toronto and London: McGraw-Hill Book Company. Available at: https://authors.library.caltech.edu/43491/10/Volume%203.pdf [Accessed: 20 August 2022].

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Nadir, A., Khan, A. & Kalim, M. 2014. Integral transforms of the generalized Mittag-Lefflerfunction. Applied Mathematical Science, 8(103), pp.5145-5154. Available at: https://doi.org/10.12988/ams.2014.43218.

Prabhakar, T.R. 1971. A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Mathematical Journal, 19(1), pp.7-15 [online]. Available at: https://ynu.repo.nii.ac.jp/?action=pages_view_main&active_a ction=repository_view_main_item_detail&item_id=6514&item_no=1&page_id=15 &block_id=22 [Accessed: 20 August 2022].

ro I

en

s± p

co c

o +-'

o

<u p

CD

i

~o <u .N "(0

CD c cd

OT OT £Z

o

>

£Z

m

co ot cd

co

O

co

cd

00

IT co X

Prajapati, J.C., Jana, R.K., Saxena, R.K. & Shukla, A.K. 2013. Some results on the generalized Mittag-Leffler function operator. Journal of Inequalities and Applications, 2013(art.number:33). Available at: https://doi.org/10.1186/1029-242X-2013-33.

> Prajapati, J.C. & Shukla, A.K., 2012. Decomposition of Generalized Mittag-

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LU

Salim, T.O. & Faraj, A.W. 2012. A generalization of Mittag-Leffler function and o£ integral operator associated with fractional calculus. Journal of Fractional Calculus ^ and Applications, 3(5), pp.1-13 [online]. Available at:

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Singh, D.K. & Rawat, R.A.H.U.L. 2013. Integrals involving generalized Mittag-Leffler function. Journal of Fractional Calculus and Applications, 4(2), pp.234-244 [online]. Available at: http://math-frac.org/Journals/JFCA/Vol4(2)_July_2013/Vol4 (2)_Papers/07_Vol.%204(2)%20July%202013, %20No.%207,%20pp.%20234-2 44..pdf [Accessed: 20 August 2022].

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Srivastava, H.M. & Tomovski, Z. 2009. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Applied Mathematics and Computation, 211(1), pp.198-210. Available at: https://doi.org/10.1016/j.amc.2009.01.055.

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o> £±

<U

НЕКОТОРЫЕ ИНТЕГРАЛЫ С ОБОБЩЕННЫМИ ^

ФУНКЦИЯМИ ТИПА МИТТАГ-ЛЕФФЛЕРА ®

б <и

Сиразул Хака, Мэгги Афанб,

Мохаммад Саид Кханб, Никола Фабианов го

С

а Дж.С. Университет Шикохабад, департамент прикладных наук, г. Фирозабад, штат Уттар-Прадеш, Республика Индия

б Университет медицинских наук Сефако Макгато, кафедра Ц

математики и прикладной математики, ®

Га-Ранкува, Южная Африка &

в Белградский университет, Институт ядерных исследований «Винча» - Институт государственного значения для Республики Сербия, г Белград, Республика Сербия, корреспондент

2 <u

РУБРИКА ГРНТИ: 27.23.17 Дифференциальное и

интегральное исчисление 27.23.21 Интегральные преобразования.

Операционное исчисление 27.23.25 Специальные функции ф

ВИД СТАТЬИ: оригинальная научная статья

"го

Резюме: ф

от

Введение/цель: В данной статье установлены опреде- ^ ленные интегралы, включающие обобщенную функцию х Миттага-Леффлера с различными типами многочленов.

Методы: Свойства обобщенной функции Миттаг-Леффлера используются в сочетании с различными видами многочленов, такими как Якоби, Лежандр и Эрмит для оценки их интегралов.

Результаты: Получены некоторые интегральные формулы, включающие функцию Лежандра, функцию Бесселя Мейтланда и обобщенные гипергеометрические функции.

Выводы: Полученные результаты носят общий характер и могут быть полезны для установления дальнейших ин-

ф

о

тегральных формул, включающих другие виды многочленов.

Ключевые слова: функция Миттаг-Леффлера, обоб- щен-ная гипергеометрическая функция, функция Бесселя-Мейтленда , многочлены Якоби, многочлены Эрмита.

гм гм о гм

Ей НЕКИ ИНТЕГРАЛИ ^И УК^УЧУJУ ГЕНЕРАЛИЗОВАНЕ

к МИТАГ-ЛЕФЛЕРОВЕ ФУНКЦШЕ

о Сиразул Хака, Маги Афанб, Мохамед Саид Канб, Никола Фабианов

< а Универзитет и.С. Шикохабад, Оде^е^е за приме^ене науке,

=5 Фирозабад, У.П., Република Инди]а

^ б Универзитет здравствених наука Сефако Макгато,

ш Департман за математику и приме^ену математику,

Га-Ранкува, Република иужна Африка

сс

<С в Универзитет у Београду, Институт за нуклеарне науке „Винча" -

Институт од националног знача]а за Републику Срби]у, Београд, Република Срби]а, ауторза преписку

ОБЛАСТ: математика ^ КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад

О

Сажетак:

>о

Увод/цил>: Дефинисани су неки интеграли ко}и укл>учу}у ге-ш нерализовану Митаг-Лефлерову функцщу са различитим

о врстама полинома.

о Методе: Сво}ства генерализоване Митаг-Лефлерове

функци]е користе се на различитим врстама полинома, као што су Jакобиjеви, Лежандрови, Ермитови, како би одредили ъихове интеграле.

Резултати: Изведене су неке интегралне формуле ко}е укя>учу]у Лежандрову функци]у, Бесел-Ме}тландову функци-}у и генерализоване хипергеометри]ске функци]е.

Закъучак: Доби}ени резултати су опште природе и могли би бити корисни за утвр^иваъе других интегралних формула ко}е укл>учу]у друге врсте полинома.

Къучне речи: Митаг-Лефлерова функци]а, генерализована хипергеометри]ска функци]а, Бесел-Ме]тландова функци-}а, Jакобиjеви полиноми, Ермитови полиноми.

EDITORIAL NOTE: The fourth author of this article, Nicola Fabiano, is a current member

oo

of the Editorial Board of the Military Technical Courier. Therefore, the Editorial Team has i ensured that the double blind reviewing process was even more transparent and more rigor- ® ous. The Team made additional effort to maintain the integrity of the review and to minimize g. any bias by having another associate editor handle the review procedure independently of the editor - author in a completely transparent process. The Editorial Team has taken special care that the referee did not recognize the author's identity, thus avoiding the conflict of interest.

КОММЕНТАРИЙ РЕДКОЛЛЕГИИ: Ччетвертый автор данной статьи Никола Фабиано & является действующим членом редколлегии журнала «Военно-технический вестник». Поэтому редколлегия провела более открытое и более строгое двойное слепое рецензирование. Редколлегия приложила дополнительные усилия для того чтобы сохранить целостность рецензирования и свести к минимуму предвзятость, ст вследствие чего второй редактор-сотрудник управлял процессом рецензирования & независимо от редактора-автора, таким образом процесс рецензирования был абсолютно прозрачным. Редколлегия во избежание конфликта интересов позаботилась о том, чтобы рецензент не узнал кто является автором статьи. Ц

РЕДАКЦШСКИ КОМЕНТАР: Четврти аутор овогчланка Никола Фабиано ]е актуелни члан Уре^ивачког одбора Во]нотехничког гласника. Због тога ]е уредништво ст спровело транспарентни|и и ригорозни|и двоструко слепи процес рецензи]е. Уложило ]е додатни напор да одржи интегритет рецензи]е и необ]ективност сведе на на]ма^у могу^у мерутако штсфдругиуредниксарадникводио процедуру рецензи]е независно од уредника аутора, при чему ]е та] процес био апсолутнотранспарентан. Уредништво ]е посебно водило рачуна да рецензент не препозна ко ]е написао рад и да не до^е до конфликта интереса.

Paper received on / Дата получения работы / Датум приема чланка: 22.08.2022 Manuscript corrections submitted on / Дата получения исправленной версии работы / q Датум достав^а^а исправки рукописа: 10.10.2022 Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 12.10.2022. ^

© 2022 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier it

(http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed under x the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).

© 2022 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons" (http://creativecommons.org/licenses/by/3.0/rs/).

© 2022 Аутори. Об]авио Во]нотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, httpV/втг.мо.упр.срб). Ово ]е чланак отвореног приступа и дистрибуира се у складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).