# SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

Embed Size (px)

### Text of SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

1/15

Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 303317http://dx.doi.org/10.4134/CKMS.2013.28.2.303

SOME PROPERTIES OF GENERALIZED

HYPERGEOMETRIC FUNCTION

Snehal B. Rao, Amit D. Patel, Jyotindra C. Prajapati,

and Ajay K. Shukla

Abstract. In present paper, we obtain functions R t(c,,a,b) and R t(c,,a,b) by using generalized hypergeometric function. A recurrence re-lation, integral representation of the generalized hypergeometric function

2R1(a, b; c; ; z) and some special cases have also been discussed.

1. Introduction and preliminaries

The special functions play very important role, particularly the hypergeo-metric function in solving numerous problems of mathematical physics, engi-neering and applied mathematics, is well-known (, , , ). This facthas inspired many mathematicians for investigations of several generalizationsof hypergeometric function (, , , , , , ).

The Gauss hypergeometric function is defined  as,

(1) 2F1(a, b; c; z) =

k=0

(a)k(b)k(c)kk!

zk (|z| < 1, c = 0, 1, 2, . . .)

and the generalized hypergeometric function, in a classical sense, has beendefined  by

pFq

a1, . . . , ap; z

b1, . . . , bq

= pFq [a1, . . . , ap; b1, . . . , bq; z](2)

=

k=0

(a1)k . . . (ap)k(b1)k . . . (bq)k

zk

k!(p = q+ 1, |z| < 1)

and no denominator parameter equals zero or negative integer.

Received May 9, 2012; Revised December 26, 2012.2010 Mathematics Subject Classification. Primary 33C20, 33E20, 26A33.Key words and phrases. generalized hypergeometric function, recurrence relation, integral

representation, fractional integral and differential operators.

c2013 The Korean Mathematical Society

303

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

2/15

304 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

E. Wright  has further extended the generalization of the hypergeometricseries in the following form:

(3) pq(z) =n=0

(1 + 1n) . . . (p + pn)

(1 + 1n) . . . (q + qn)

zn

n!,

where r and t are real positive numbers such that 1+q

t=1 tp

r=1 r > 0.When r and t are equal to 1, Equation (3) differs from the generalized

hypergeometric function pFq by a constant multiplier only. This generalizedform of the hypergeometric function has been investigated by M. Dotsenko ,V. Malovichko  and others. One of the interesting special case considered in has the following form:

2R,1 (z) = 2R1 (a, b; c; , ; z) =

(c)

(a) (b)

n=0

(a + n)

b +

n

c +

n

zn

n!;(4)

( (a) > 0, (b) > 0, (c) > 0) .

Here , both either positive or negative simultaneously, |z| < 1.The function 2R

,1 (z) is not symmetric with respect to the parameters a

and b. By letting

= > 0 in Equation (4), Virchenko et al.  defined the

generalized hypergeometric function in a different sense as:

(5) 2R1 (z) = 2R1 (a, b; c; ; z) =

(c)

(b)

k=0

(a)k (b + k)

(c + k) k!zk; > 0, |z| < 1.

If = 1, then (5) reduces to a Gausss hypergeometric function 2F1(a, b; c; z).

For > 0, on |z| = 1, the function 2R1 (a, b; c; ; z) is defined provided (c a b) > 0, as discussed in the proposition of Appendix A.Rao et al. ,  studied various properties of generalized hypergeometric

function in the light of fractional calculus.The Riemann-Liouville fractional integral of order is defined as :For () > 0,

(6) If (t) =1

()

t0

(t )1 f() d

and the fractional differential operator of order defined as :

(7) Df(t) = Dn

Inf(t)

,

where () > 0, and n is the smallest integer with the property that n > .The Laplace transform of the function f(z) is defined as :

(8) L {f(z)} =

0

eszf(z) dz.

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

3/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 305

2. Fractional op erators and the generalized hypergeometricfunction 2R1 (a, b; c; ; z)

Consider the function f(t) = 1(b)

k=0(a)

k(b+k)(ct)k

(k!)2= 2F1 (a, b; 1; ct),

where a, b C ( (a) > 0, (b) > 0) and c is arbitrary constant such that|ct| < 1.

On applying the fractional integral operator (6) of order on f(t), we give

If(t) =1

()

t0

(t )

1f()

d

=1

()

t0

(t )

1 1

(b)

k=0

(a)k (b + k) (c)k

(k!)2

d

=t

(+ 1) (+ 1)

(b)

k=0

(a)k (b + k) (ct)k

(+ 1 + k) k! ,

one can easily write this in following form:

(9)t

(+ 1)2R1 (a, b; + 1;1; ct) =

t

(+ 1)2F1 (a, b; + 1; ct) .

Here we denote (9) as Rt (c,,a,b), i.e.,

Rt (c,,a,b) =t

(+ 1)2R1 (a, b; + 1;1; ct)(10)

=t

(+ 1)

2F1 (a, b; + 1; ct) .

Now, applying the fractional differential operator (7) of order on f(t), weget

Df(t) =

d

dt

n In

1

(b)

k=0

(a)k (b + k) (ct)k

(k!)2

= Dn

tn

(b)

k=0

(a)k (b + k) (ct)k

(1 + n + k) k!

which yields

Df(t) = t

(1 )2R1 (a, b; 1 ; 1; ct)(11)

=t

(1 )2F1 (a, b; 1 ; ct) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

4/15

306 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

On denoting (11) as

Rt (c, ,a,b) = t

(1 )2R1 (a, b; 1 ; 1; ct)(12)

=t

(1 )2F1 (a, b; 1 ; ct) .

3. Properties of the functions Rt (c , , a , b) and Rt (c, , a , b)

Theorem 3.1. If a, b C ( (a) > 0, (b) > 0) and c is arbitrary constantsuch that |ct| < 1, then

(13) IRt (c,,a,b) = Rt (c, + ,a,b) ,

(14) DRt (c,,a,b) = Rt (c, ,a,b) .

For n N and as any constant;The Laplace transform of Rt (c,, n, + n 1) is given as

(15) L {Rt (c,, n, + n 1)} =1

s+1 yn (c; , s) ,

where yn (c; , s) is the generalized Bessel polynomial .

Proof. From (6) and the left-hand side of (13), we get

I

Rt (c,,a,b) =

1

()t

0 (t )

1

R (c,,a,b) d

=1

()

t0

(t )1

(+ 1)2R1 (a, b; + 1;1; c)

d

=1

()

t0

(t )1

(b)

k=0

(a)k (b + k) (c)k

( + 1 + k) k!

d

which gives

I Rt (c,,a,b)

=1

() 1

0

(1 x)1

t1 (xt)

(b)

k=0

(a)k (b + k) (cxt)k

( + 1 + k) k! t dx=

t+

( + + 1)2R1 (a, b; + + 1; 1; ct) = Rt (c, + ,a,b) .

This is the proof of (13).

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

5/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 307

From (7) and left hand side of (14);

D Rt (c,,a,b) = Dn InRt (c,,a,b)

= Dn [Rt (c, n + ,a,b)]

= Dn

tn+

(n + + 1)2R1 (a, b; n + + 1;1; ct)

=t

( + 1)2R1 (a, b; + 1;1; ct)

= Rt (c, ,a,b)

which is (14).On setting n N, replacing a by n and b by + n 1 in Rt (c,,a,b),

where as any constant and then taking Laplace transform of (10), it yields

L {Rt (c,, n, + n 1)}

= L

t

(+ 1)2R1 (n, + n 1; + 1; 1; ct)

=

0

est

t

(+ 1)2R1 (n, + n 1; + 1; 1; ct)

dt

=1

s+1

nk=0

(n)k ( + n 1)kk!

cs

k

=1

s+12F0

n, + n 1; ;

c

s

=1

s+1 yn (c; , s) .

Thus, the Laplace transform of Rt (c,, n, + n 1) is

L {Rt (c,, n, + n 1)} =1

s+1 yn (c; , s) ,

where yn (c; , s) is the generalized Bessel polynomial .This proves (15).

Theorem 3.2. Let a, b C ( (a) > 0, (b) > 0) and c is arbitrary constantsuch that |ct| < 1, () < 1. Then

IRt (c, ,a,b) = Rt (c, ,a,b) ,(16)

DRt (c, ,a,b) = Rt (c, ,a,b) .(17)

For n N and as any constant;

The Laplace transform of Rt (c, , n, + n 1) is given as

(18) L {Rt (c, , n, + n 1)} =1

s1 yn (c; , s) ,

where yn (c; , s) is the generalized Bessel polynomial .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

6/15

308 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

Proof. From (6) and left hand side of (16), we get

IRt (c, ,a,b) =1

()t

0(t )

1

R (c, ,a,b) d

=1

()

t0

(t )1

(b)

k=0

(a)k (b + k) (c)k

(1 + k) k!

d

which, upon substituting = xt, yields

IRt (c, ,a,b)

=1

()

10

(1 x)1

t1

(xt)

(b)

k=0

(a)k (b + k) (cxt)k

(1 + k) k!

t dx

=t

( + 1)2R1 (a, b; + 1; 1; ct) = Rt (c, ,a,b) .

This is the proof of (16).From (7) and left hand side of (17), we get

DRt (c, ,a,b)

= Dn

InRt (c, ,a,b)

= Dn [Rt (c, n ,a,b)]

= Dn

tn

(n + 1)2R1 (a, b; n + 1; 1; ct)

=t

( + 1)

( + 1)

(b)

k=0

(a)k (b + k)

( + 1 + k)

(ct)k

k!

= t

( + 1)2R1 (a, b; + 1;1; ct) ,

this can also be written as Rt (c, ,a,b) . This leads to (17).On setting n N, replacing a by n and b by + n 1 in Rt (c, ,a,b),

where as any constant and then taking Laplace transform of (12), it yields

L {Rt (c, , n, + n 1)}

= L

t

(1 )2R1 (n, + n 1; 1 ; 1; ct)

=

0

est

t

(1 )2R1 (n, + n 1; 1 ; 1; ct)

dt

= 1s+1

nk=0

(n)k ( + n 1)kk!

cs

k

=1

s12F0

n, + n 1; ;

c

s

=

1

s1 yn (c; , s) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

7/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 309

Thus, the Laplace transform of Rt (c, , n, + n 1) is given as

L {Rt (c, , n, + n 1)} =

1

s1 yn (c; , s) ,where yn (c; , s) is the generalized Bessel polynomial .

This is the proof of (18).

4. Recurrence relation for the generalized hypergeometric function

2R1 (a, b; c; ; z)

Theorem 4.1. If (k) > 0, (a) > 0, (b) > 0, (m) > 0, |z| < 1, then

(m + 1) 2R1 (a, b; c + s + 1; k; z) 2R1 (a, b; m + 2; k; z)(19)

=

(k)2

(m + 2)

z22R1 (a, b; m + 3; k; z) +

(k)

(m + 2){(k) + 2 (m + 1)} z

2R1 (a, b; m + 3; k; z) + (m) 2R1 (a, b; m + 3; k; z) ,

where 2R1 (a, b; c; ; z) =ddz 2R1 (a, b; c; ; z) and 2R1 (a, b; c; ; z) =

d2

dz2 2R1 (a,b; c; ; z).

Proof. On applying the fundamental relation of the Gamma function (z + 1)= z (z) to (5), we can write

(20) 2R1 (a, b; m + 1; k; z) = (m + 1)

(b)

n=0

(a)n (b + kn)

(m + kn) (m + kn)

zn

n!

and(21)

2R

1(a, b; m + 2; k; z) =

(m + 2)

(b)

n=0

(a)n (b + kn)

(m + 1 + kn) (m + kn) (m + kn)

zn

n!.

On writing equation (21) as:

2R1 (a, b; m + 2; k; z)

(22)

= (m + 2)

(b)

n=0

1

(m + kn)

1

(m + 1 + kn)

(a)n (b + kn)

(m + kn)

zn

n!

= (m + 1) 2R1 (a, b; m + 1; k; z) (m + 2)

(b)

n=0

(a)n (b + kn)

(m + 1 + kn) (m + kn)

zn

n!.

For our convenience, we, denote the last summation in (22) by S as:

S = (m + 2)

(b)

n=0

(a)n (b + kn)

(m + 1 + kn) (m + kn)

zn

n!(23)

= (m + 1) 2R1 (a, b; m + 1; k; z) 2R1 (a, b; m + 2; k; z) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

8/15

310 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

On applying a simple identity 1u

= 1u(u+1) +

1u+1

, (u = kn + m + 1) to (23), we

obtain

S = (m + 2)

(b)

n=0

(m + kn) (a)n (b + kn)

(m + 3 + kn)

zn

n!

+ (m + 2)

(b)

n=0

(c + s + kn) (m + 1 + kn) (a)n (b + kn)

(m + 3 + kn)

zn

n!

and

S =k

(m + 2)

(m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!

(24)

+m

(m + 2) (m + 3)

(b)

n=0

(a)n (b + kn)

(m + 3 + kn)

zn

n! +

k2

(m + 2)

(m + 3)

(b)

n=1

n (a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!

+

(m + 2)

(m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!

+

(m + 2)

(m + 3)

(b)

n=0

(a)n (b + kn)

(m + 3 + kn)

zn

n!

,

where = k (2m + 1) and = m (m + 1).We now express each summation in the right hand side of (24) as follows:

d2dz2

z22R1 (a, b; m + 3; k; z)

(25)

= 2 2R1 (a, b; m + 3; k; z) + 4z 2R1 (a, b; m + 3; k; z)

+ z2 2R1 (a, b; m + 3; k; z)

and

d2

dz2

z22R1 (a, b; m + 3; k; z)

(26)

= (m + 3)

(b)

n=0

(n + 2) (n + 1) (a)n (b + kn)

(m + 3 + kn)

zn

n!

=

(m + 3)

(b)

n=1

n (a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!

+ 3 (m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!+ 2 2R1 (a, b; m + 3; k; z) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

9/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 311

(25) and (26) imply

(m + 3) (b)

n=1

n (a)n (b + kn) (m + 3 + kn)

zn

(n 1)!(27)

= z22R1 (a, b; m + 3; k; z) + 4z 2R1 (a, b; m + 3; k; z)

3 (m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!.

Let

d

dz(z 2R1 (a, b; m + 3; k; z))(28)

= 2R1 (a, b; m + 3; k; z) + z 2R1 (a, b; m + 3; k; z)

andd

dz(z 2R1 (a, b; m + 3; k; z))(29)

= (m + 3)

(b)

n=0

(n + 1) (a)n (b + kn)

(m + 3 + kn)

zn

n!

= (m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!+ 2R1 (a, b; m + 3; k; z) .

From (28) and (29), we get

(30) (m + 3)

(b)

n=1

(a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!= z

2R

1(a, b; m + 3; k; z) .

Combining (27) and (30), it yields

(m + 3)

(b)

n=1

n (a)n (b + kn)

(m + 3 + kn)

zn

(n 1)!(31)

= z22R1 (a, b; m + 3; k; z) + z2R1 (a, b; m + 3; k; z) .

Now applying (30) and (31) to (24), we get

S =

k2

(m + 2)

z22R1 (a, b; m + 3; k; z)(32)

+

k2 + k + (m + 2)

z2R1 (a, b; m + 3; k; z)

+(m + )

(m + 2)2R1 (a, b; m + 3; k; z) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

10/15

312 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

From (22), (23) and (32), we arrive at

(m + 1) 2R1 (a, b; m + 1; k; z) 2R1 (a, b; m + 2; k; z)

=

k2

(m + 2)

z22R1 (a, b; m + 3; k; z) +

k

(m + 2){k + 2 (m + 1)}

z2R1 (a, b; m + 3; k; z) + m2R1 (a, b; m + 3; k; z) .

5. Some integral representations of the generalized hypergeometricfunction

Theorem 5.1. For k > 0, (a) > 0, (b) > 0, (m) > 0, (m a b) > 0we get

(33)

10

tm2R1

a, b; m; k; tk

dt =2R1 (a, b; m + 1; k; 1)

(m) 2R1 (a, b; m + 2; k; 1)

(m) (m + 1) .

Proof. On putting z = 1 in (23), it yields

(m)

(b)

n=0

(a)n (b + kn)

(m + 1 + kn) (m + kn) n!(34)

=2R1 (a, b; m + 1; k; 1)

m

2R1 (a, b; m + 2; k; 1)

(m + 1) m.

Let

z

0

tm2R1 a, b; m; k; tk dt = z

0

tm (m)

(b)

n=0

(a)n (b + kn)

(m + kn)

tkn

n! dt(35)=

(m)

(b)

n=0

(a)n (b + kn) zm+1+kn

(m + 1 + kn) (m + kn) n!.

Now, comparing (34) and (35) after setting z = 1 in (35), we have10

tm2R1

a, b; m; k; tk

dt =2R1 (a, b; m + 1; k; 1)

m

2R1 (a, b; m + 2; k; 1)

(m) (m + 1).

This is the proof of Theorem 5.1.

Theorem 5.2. If a,b,c, C such that (a) > 0, (b) > 0, (c) > 0, (c) > 0, k > 0 and |z| < 1, then

2R1 (a, b; c; ; z)

= kzc

0

exp

tk

zk

tc1

n=0

(b + n) (c) (a)n tn

(b) (c + n) c+n

k

dt.

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

11/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 313

Proof. Consider,

0exp

t

k

zktc1

n=0(b+n)(c)(a)

ntn

(b)(c+n)( c+nk )

dt and su-

bstituting tk

zk = u, we get

=

0

euzc1uc1

k

n=0

(b + n) (c) znun

k (a)n

(b) (c + n) n!

c + n

k

zk u 1kk du

=zc

k

n=0

(b + n) (c) zn (a)n (b) (c + n) n!

c+n

k

c + nk

which, on further simplification, gives

0

exptk

z

ktc1

n=0

(b + n) (c) (a)n tn

(b) (c + n) c+n

k dt

=zc

k2R1 (a, b; c; ; z) .

This is the proof of Theorem 5.2.

Theorem 5.3. If a,b,c, C such that (a) > 0, (b) > 0, (c) > 0, (c) > 0 and |z| < 1. Then

2R1 (a, b; c; ; z) = (c)

() (c )

10

1 t 1

c1

2R1 (a, b; ; ; tz) dt.

Proof. Let 10

1 t

1

c12R1 (a, b; ; ; tz) dt

=

10

1 t 1

c1 n=0

(a)n (b + n) () (tz)n

(b) ( + n) n!

dt.

On substituting t1 = u, we get

10

1 t 1

c1

2R1 (a, b; ; ; tz) dt

=

n=0

(a)n (b + n) () (z)n

(b) ( + n) n! (c , + n)

= () (c )

(c)2R1 (a, b; c; ; z) .

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

12/15

314 S. B. RAO, A. D. PATEL, J. C. PRAJAPATI, AND A. K. SHUKLA

This can easily be written as

2R1 (a, b; c; ; z) =

(c)

() (c )1

0

1 t

1c1

2R1 (a, b; ; ; tz) dt.

This is the proof of Theorem 5.3.

Theorem 5.4. If a,b,c, C such that (a) > 0, (b) > 0, (c) > 0, (c) > 0 and |z| < 1, then

2R1 (a, b; c; ; z)

= (c)

() (c )

10

t1 (1 t)c1 2R1 (a, b; c ; ; z (1 t)

) dt.

Proof.

1

0

t1 (1 t)c12R1 (a, b; c ; ; z (1 t)

) dt

=

10

t1 (1 t)c1

(c )

(b)

n=0

(a)n (b + n)

(c + n)

zn (1 t)n

n!

dt

= (c )

(b)

n=0

(a)n (b + n)

(c + n)

zn

n!

10

t1 (1 t)c+n1

dt

= () (c )

(c)

(c)

(b)

n=0

(a)n (b + n)

(c + n)

zn

n!

= () (c )

(c)2R1 (a, b; c; ; z) .

This gives

2R1 (a, b; c; ; z)

= (c)

() (c )

10

t1 (1 t)c1

2R1(a, b; c ; ; z (1 t)

)dt,

which proves Theorem 5.4.

Appendix: A

Proposition. For > 0, and (a) > 0, (b) > 0, (c) > 0; on |z| = 1, thefunction 2R1 (a, b; c; ; z) is defined provided (c a b) > 0.

Proof. For > 0, and (a) > 0, (b) > 0, (c) > 0 the function 2R1(a, b; c; ;z) is

(36) 2R1 (z) = 2R1 (a, b; c; ; z) =

(c) (b)

n=0

(a)n (b + n) (c + n) n!

zn.

We have to prove its convergence condition for |z| = 1 by using the comparisontest.

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

13/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 315

For |z| = 1,

n=0 un =

n=0

(c)(b)

(a)n

(b+n)(c+n)n!

zn

= 1 +

n=1

(c)(b)

(a)n

(b+n)(c+n)n!

and for (c a b) > 0, n=1 vn = n=1 1

n

1+ ; = 12

(c a b) > 0,limn

un

vn

= limn

(c) (b) (a)n (b + n) (c + n) n!

1

n1+

= limn

n1+ (c) (b) (a)n (b + n) (c + n) n!

= limn

(a)n(n 1)!na (b + n) (b) (n 1)!nb (c) (n 1)!nc

(c + n)

(n 1)!n1+

n!ncab

= lim

n 1

(a)

(b + n)

(b) (n 1)!nb (c) (n 1)!nc

(c + n) limn 1

ncab = (Finite Number) lim

n

1ncab (Explanation is given below in Section I)

= 0, because (c a b ) = 2 > 0.

Therefore

n=0 un= 1+

n=1

(c)(b) (a)n(b+n)(c+n)n! is convergent, since n=1 vnis convergent.

Thus the series in (36) is absolutely convergent on |z| = 1 when (c a b)> 0.

Hence, 2R1 (z) = 2R1 (a, b; c; ; z) =

(c)(b)

n=0(a)

n(b+n)

(c+n)n! zk is convergent

for |z| = 1 when (c a b) > 0.

Section I : For (a) > 0, n N, 1, 0 < | (a + n)| | (a + n)|. There-

fore, 0 < 1(a+n) 1(a+n) .

Thus

0 < limn

(n 1)!na (a + n) limn

(n 1)!na (a + n) 0 < limn

(n 1)!na (a + n) 1.

(37)

Also, for (a) > 0, n N, 0 < 1, 0 < | (a + n)| | (a + n)|.Which implies that

0 0.

References

 L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman andHall/CRC press, Boca Raton, FL, 2007.

 M. Dotsenko, On some applications of Wrights hypergeometric function, C. R. Acad.Bulgare Sci. 44 (1991), no. 6, 1316.

 A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher TranscendentalFunctions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London,1953.

 A. A. Kilbas and M. Saigo, H-Transforms, Chapman and Hall/CRC press, Boca Raton,FL, 2004

 A. A. Kilbas, M. Saigo, and J. J. Trujillo, On the generalized Wright function, Fract.Calc. Appl. Anal. 5 (2002), no. 4, 437460.

 V. Kiryakova, Generalized Fractional Calculus and Applications, Wiley & Sons. Inc.,New York, 1994.

 H. L. Krall and O. Frink, A new class of orthogonal polynomials: The Bessel polynomials,Trans. Amer. Math. Soc. 65 (1949), 100115.

 V. Malovichko, A generalized hypergeometric function, and some integral operators thatcontain it, Mat. Fiz. Vyp. 19 (1976), 99103.

 A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions With Applica-tions in Statistics and Physical Sciences, Springer-Verlag, Berlin, 1973.

 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and FractionalDifferential Equations, John Wiley & Sons, Inc., 1993.

 R. K. Raina, On generalized Wrights hypergeometric functions and fractional calculusoperators, East Asian Math. J. 21 (2005), no. 2, 191203.

 E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960. S. B. Rao, J. C. Prajapati, A. D. Patel, and A. K. Shukla, On generalized hypergeometric

function and fractional calculus, Communicated for publication. S. B. Rao, J. C. Pra japati, and A. K. Shukla, Generalized hypergeometric function and

its properties, Communicated for publication. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives:

Theory and Applications, Gordon and Breach Science publishers, Yverdon (Switzer-land), 1993.

 N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract.Calc. Appl. Anal. 2 (1999), no. 3, 233244.

 , On the generalized confluent hypergeometric function and its applications,Fract. Calc. Appl. Anal. 9 (2006), no. 2, 101108.

 N. Virchenko, S. L. Kalla, and A. Al-Zamel, Some results on a generalized hypergeomet-ric function, Integral Transform. Spec. Funct. 12 (2001), no. 1, 89100.

 N. Virchenko, O. Lisetska, and S. L. Kalla, On some fractional integral operators in-volving generalized Gauss hypergeometric functions, Appl. Appl. Math. 5 (2010), no.10, 14181427.

• 7/30/2019 SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

15/15

SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION 317

 N. Virchenko and Olena V. Rumiantseva, On the generalized associated Legendre func-tions, Fract. Calc. Appl. Anal. 11 (2008), no. 2, 175185.

 E. M. Wright, On the coefficient of the power series having exponential singularities , J.London Math. Soc. 8 (1933), 7179.

Snehal B. Rao

Department of Applied Mathematics

The M.S. University of Baroda

Amit D. Patel

Department of Applied Mathematics and Humanities

S.V. National Institute of Technology

Surat-395 007, India

Jyotindra C. Prajapati

Department of MathematicsCharotar Institute of Technology

Changa, Anand-380 421, India

Ajay K. Shukla

Department of Applied Mathematics and Humanities

S.V. National Institute of Technology

Surat-395 007, India ##### Ranjan K. Jana, Bhumika Maheshwari and Ajay K. …...Note on extended hypergeometric function 593 2. Generalized integral transforms In 2012, Virchenko  introduced the following
Documents ##### Fractional integral and generalized Stieltjes transforms ...Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators Tom Koornwinder
Documents ##### Hypergeometric3F2 - FunctionHypergeometric3F2 Notations Traditional name Generalized hypergeometric function 3F2 Traditional notation 3F2Ha1,a2,a3;b1,b2;zL Mathematica StandardForm
Documents ##### Generalized switching function model of modular multilevel ... · PDF fileGeneralized switching function model of modular multilevel converter ... paper presents a generalized switching
Documents ##### Some properties of a hypergeometric function which appear ...gvm/radovi/GVM-MTR-JGO.pdf · Some properties of a hypergeometric function which appear in an approximation problem Gradimir
Documents ##### Padé approximations of generalized …cc.oulu.fi/~tma/TOKYOSLIDES.pdfAbstract We shall present short proofs for type II Pad e approximations of the generalized hypergeometric and
Documents ##### Transformation formulas for the generalized hypergeometric ......the generalized hypergeometric functions r+2F r+1(x) and r+1F r+1(x), where r pairs of numeratorial and denominatorial
Documents ##### Hypergeometric Functions: From One Scalar …Zonal Polynomials approach 4.1.3. Matrixtransforms approach- 4.2. Hypergeometric function in two matrix variates 5. Computational Issues
Documents ##### Multiplication and Translation Formulas for the Generalized Hypergeometric … · 2019-11-20 · Index Terms—Generalized Hypergeometric Function, Basic Hypergeometric Series, Classical
Documents ##### Towards all-order Laurent expansion of generalized ... all-order Laurent expansion of generalized hypergeometric functions around rational values of ... ···Sap(j−1), ... of the
Documents ##### New results on asymptotics of holonomic sequencespoulalho/ALEA09/slides/banderier.pdf · Young tableaux of bounded height (generalized) hypergeometric functions Latin squares the
Documents ##### François Cassier fcassier@nag - RWTH Aachen University...Numerical Excellence Hypergeometric function Robust accurate real confluent hypergeometric function Nearest correlation matrix
Documents ##### The Möbius function of generalized subword order › fpsac12 › download › slide › 0730 › ... · 2012-08-03 · The Möbius function of generalized subword order Peter McNamara
Documents ##### THREE LECTURES ON HYPERGEOMETRIC FUNCTIONSpeople.math.umass.edu/~cattani/hypergeom_lectures.pdf · THREE LECTURES ON HYPERGEOMETRIC FUNCTIONS ... The Gamma Function and the Pochhammer
Documents ##### Applications of the Hypergeometric Method to the ...bennett/BaB.pdf · Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation ... as we shall see in
Documents ##### The Application of Generalized Prolate Spheroidal Wave ......multivariable H-function in heat conduction. In this paper we employ the generalized prolate spheroidal wave function,
Documents ##### Exact Graetz problem solution by using hypergeometric function
Documents ##### LAGUERRE FUNCTIONS ON SYMMETRIC CONES AND … · Generalized hypergeometric functions are extended to homogeneous cones. Many important di erential properties also extend. Around
Documents ##### GENERALIZED DEDEKIND ETA-FUNCTIONS AND …...Key words and phrases. Eisenstein series, Dedekind eta-function, generalized Dede-kind eta-function, Dedekind sum, generalized Dedekind
Documents ##### Multiple integrals transformation about the …...Multiple integrals transformation about the generalized incomplete hypergeometric function,a general class of polynomials and the
Documents ##### Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function
Design ##### The Open Statistics & Probability Journal · Bayesian Inference for Three The Open Statistics & Probability Journal, 2017, Volume 08 29. generalized hypergeometric function, denoted
Documents Documents