Certain sums of double hypergeometric series

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

R. P. S INGAL

Certain sums of double hypergeometric series

Rendiconti del Seminario Matematico della Università di Padova, tome 41 (1968), p. 227-234

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HYPERGEOMETRIC SERIES

R. P. SINGAL

*)

1. Let

Kamp6

^{de F6ri6t function}

[1]

^{in a} modified notation be defined as

where stands for the sequence

ai , a2 ~ ... , a~ ;

and

(a)m

⁼

a (a + 1 ) ... (ac +

^{m -}

1 ), (a)o

^{= 1.}

Carlitz

[3-6]

^has

given

^{four sums of}

F11,’12

^{and a sum}

under different sets of condition. Jain

[7]

^has

given

^{a sum of}

where a

-~-

^{b - c is not an}

integer.

Recently

Srivastava

~) [9]

^{studied the sum of}

Fl’12

^{in a more}

systematic

way and gave ^a number of results.

~) ^{Indirizzo dell’A.} Depont. ^{of Math.} Thapar Institute of Enginnering ^and Technology, ^Patiala (India).

1) ^{I am thankful to} prof. ^G. P. Srivastava who has so kindly ^{alluwed me}

to go through ^his unpublished manuscript

[9].

228

The

object

^of this paper is ^to

study

^{the sum of}

Fl ’1,

^{in detail.}

Now

using

^the transformation

[2]

we

get

Choosing

and

4F3

^{on the}

^right

^{reduces to}

3F2

Saalschiitzian.

Using

Saalschützian theorem

[8],

^we

get

This

4F3

^{reduces to}

2F1

under six different sets of conditions and the

resulting

^can be summed

by

Gauss theorem.

Thus

assuming

^the conditions

(2.3)

&#x26;

(2.4)

^are

satisfied,

^we

get

which is the same as

( 1.1 )

provided

^{b is not a}

positive integer

^n.

230

Now

using (2.1)

^{in the}

right

^{hand side of}

(2.2), assuming

d = b

-+-

^{b’ and}

changing

^{the order of}

summation,

^we

get

This is Saalschftzian if

(2.4)

is satisfied.

If c ⁼ b

+

^{c’ - a’}

+

n,

4F.

^{reduces to}

F2

^{which can be sum-}

med

by

Saalschutzian theorem and ^we

get

If further ^c = b

+ n -

ⁱⁿ it reduces to

2Fl

and is summed

by

Gauss theorem. Hence

Similarly

^{if we take d - n in}

(2.11)

^{and assume condition}

(2.4)

is

satisfied,

^then

summing 3F2 ^by

Saalsehiitz’s theorem we

get

This reduces to

2F1

^{if either}

which can be summed

by

Gauss theorem. Thus

provided

and

provided

^I

3. In this

section

^we consider that case of

F1°,’13

^{in which}

one

parameter

^{in the numerator cancell one}

parameter

^{in the de-} nominator.

which

by

^Gauss theorem is

choosing d

^{= b}

-t- b’,

ⁱⁿ

(3.1)

^{reduces to}

SF2.

Thus

It can be summed

by

Gauss theorem i Thus

232

Using

Saalschftz’s

theorem,

^we

get

provided

^{and a or c is a nega-}

tive

integer.

Using

^Dixon

theorem,

^we

get

provided

when d = b

+

^{b’ and}

The above ^can be summed

by

Watson theorem in four different sets of conditions. Sum under one set of condition is

The can also be summed

by Whipples

theorem in two dif- ferent sets of conditions. Sum in one case is

Taking

^{c = d -- b’ -}

c’, 4Fs

ⁱⁿ

^(3.1)

^{reduces to}

3F2

^{which can be}

summed

by Saalsehiitzs, Dixon, Whipples theorem,

^hence

giving

at least four different sums of

.F’°’i .

^*

I express my sincere thanks to Dr. S. Saran

(Punjabi

^Univer-

sity)

^{for his kind}

supervision

^{in the}

preparation

^{of this} paper.

234

REFERENCES

[1]

^{APPEL P. et KAMPÉ DE FÉRIÉT: Functions} Hypergeometriecs ^et hyperspheriques,

Gauthiar Villars, ^Paris (1926);

[2]

^{BAILEY, W. N: On the sum} of a terminating 3F2 (1): Quart. ^{J. Math.} (Oxford) (2) ⁴ (1953), 237-40 ;

[3]

CARLITZ, ^{L. : Another} Saalschützian theorem for double series : J. ^Lond Math.

Soc. Vol. 58 (1963), ^415-418.

[4]

CARLITZ, ^{L. :} Another Saalschützian theorem for ^{double series :} Rend. Semi. Mat Univ. Padova, ^{Vol. 34} (1965) 200-203 ;

[5]

CARLITZ, L. : A summation theorem for ^double Hypergeometric ^{Series Rend.}

Semi. Mat. Univ. Padova: (1967) ^230-233.

[6]

CARLITZ, ^{L.: Summation} of ^{a double} Hypergeometric Series. Matematiche (Ca- tania) ^{Vol. 22} (1967) 138-142;

[7]

JAIN, R. N.: Sum of a ^double Hypergeometric ^Series. Matematiche (Catania)

Vol. 21 (1966) 300-301;

[8]

SLATER, ^L. J. : Generalized Hypergeometric functions (Camb. Press) 1967 ;

[9]

SRIVASTAVA, ^{G. P. : Summation} of double series of Saalschützian type (to appear).

Manoscritto pervennto ⁱⁿ redazione il 8-6-68.