Some formulae by means of fractional derivatives

(1)

C ^OMPOSITIO M ^ATHEMATICA

H. L. M ^ANOCHA B. L. S ^HARMA

Some formulae by means of fractional derivatives

Compositio Mathematica, tome 18, n

^o

3 (1967), p. 229-234

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(2)

229

by

H. L. Manocha and B. L. Sharma

1

The

object

of this paper is ^toobtain some

formulae,

^believed

to be _new,

by using

^the

conception

^offractional derivatives as

defined

by

2

To start

with,

^let^us^consider^the

elementary identity

or

Now we

multiply

both the sides

by

^{Ub-1 and}^then

apply

^the

operator, Db-cu,

^so^that

Using

^thedefinition

(1)

^it^can^be

easily

^seen^that

and

(3)

230

Thus we arrive at

or,

putting u

⁼^1,

In case we

put

^z⁼

z+y-zy

^and^usethe formula

[2,

p.

239]

(2)

^reduces^to

On the other

hand,

^if^we

put z

⁼

x+y,

it follows from

(2)

^that

For A ⁼0,

(5)

^reduces^to^{the known}^result^due^to^Burchnall

and

Chaundy [1].

In order to find an inverse of

(2),

^we^consider

or

We

multiply

^both^the^sides

by

^{Ub-1 and}

apply

^the

operator Db-cu,

thus

having

(4)

which

yields,

^on

putting u

⁼^1,

Putting z

⁼

x-E-y

^and

using (3), (6)

^becomes

while for z

= x+y-xy,

^we^have

Again,

^{for A}⁼0,

(8) yields

^{the known}formula due to Burchnall and

Chaundy [1].

Similarly,

^if^weconsider the

identity

or

and

apply

^the

operator Da-cu,

^we

get

For u ⁼1, ^we

get

the formula due to Burchnall and

Chaundy [1].

3 We take up another

elementary

^result

or

(5)

232

Now we

multiply

both the sides

by x03B2-1y03B2’-1,

^so^that

Applying

^the

operators Dfl-7

^and

D03B2’-03B3’y,

^we^obtain

Using

^Euler’s^formula

we rewrite

(10)

^as

which on

putting y

⁼

y’

⁼^oc,

fl’ =03B2, 2x/(2x-1)

⁼y and

using

the relation

[2,

p.

238]

becomes

Dividing

^y

by 03B2

^and

taking P

^~^oo,^we

get

^from

(11)

^that

or, if we

prefer,

Next,

^we^consider

(6)

or

Multiplying

both the sides

by

^x03B1-1

y03B3-1

^and

applying

^the

operators D:-P

^and

Dÿ-a,

^we

get

Lastly,

^we^{deal with}^the

identity

or

Applying

^the

operators Da1-b1x

^and

D03B11-03B21y,

^it

^yields

If we continue this process of

performing operators

p times with

respect

^{to x}and 1 times with

respect

^toy and then suppress ^some

parameters,

^we^arrive^at

(7)

234

REFERENCES

BURCHNALL, J. L. and CHAUNDY, ^{T. W.}

[1] "Expansions ^ofAppell’s ^doublehypergeometric functions" Quart. ^{J. Maths.}

(Oxford), ^vol.^11,^No.^44,pp. 2492014270 (1940).

ERDELYI, ^A.

[2] Higher transcendental functions. Vol. 1 (1953).

(Oblatum 17-10-1966). Department ^ofApplied ^Sciences Punjab Engineering College Chandigarh (India) Department ^{of Maths.}

p.u. Regional ^{Centre of}

Post Graduate Studies Simla (India)

Some formulae by means of fractional derivatives

C OMPOSITIO M ATHEMATICA

H. L. M ANOCHA B. L. S HARMA