Certain properties of Meijer-Laplace transform

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C ^OMPOSITIO M ^ATHEMATICA

V. M. B ^HISE

Certain properties of Meijer-Laplace transform

Compositio Mathematica, tome 18, n

^o

1-2 (1967), p. 1-6

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1

by V. M. Bhise

1

A

generalization

of the classical

Laplace

^transform

has been introduced

by

the author

[1,

p.

57]

in the form

and denoted

symbolically by ~(p)

⁼

G[f(t);

am, nm,

a].

^Certain

rules and recurrence relations have also been established

[2]

^for

the transform

(1.2),

^which^weshall call as

Meijer-Laplace

^trans-

form.

Setting aj

⁼

0, i

⁼1, 2, ^...,^m-1.

(i)

^{and am}⁼^0,^ar⁼0,

(1.2)

^reduces^to

(1.1)

(ii)

^andam

= -m’-k, nm

⁼

m’-k,

^or^{= -m’-k}^we

get (1.2)

reduced to

Meijer’s

^transform

[3,

^p.

730],

(iii) and am

⁼

1 2-m’-k, nm

⁼

2m’, 03C3

⁼⁰^we

get (1.2)

^reduced

to Verma’s transform

[4,

p.

209]

We denote

(1.1), (1.3)

^and

(1.4) symbolically by

respectively.

In this paper ^wehave obtained a theorem for the

Meijer- Laplace

^transform

(1.2),

^{which is}^a

generalization

^{of the}^well-

known Tricomi’s theorem

[5,

p.

564].

^Various

particular

^cases

have been

given

and the theorem is illustrated

by

^an

example.

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2

THEOREM : If

then,

where 1

provided

^{n, s}^are

positive integers s

>

n; arg ys/pn| (s-n)03C0/2;

min

(Re

~j, Re

p )

+ ⁿ^min

(Re ni/s,

^Re

als)

> -1 - Re 03BB >

-

Re(~j+03B1j+03BB)-2

^{for i}⁼^{1, 2,}^...,

m; j =

1, 2,

..., /À; 1

⁼^0,

1, ^...,

n -1;

^v⁼0, 1, ^...,^{s -1.}^the

integral

ⁱⁿ

(2.1)

^is

absolutely

and

uniformly convergent

^{and the}

Meijer-Laplace

transform of

ItÀ ~0 F(t, y)dyl |

^exists.

PROOF: From

(1.2), replacing

p

by p-"’8

^and

multiplying by

pn/s-03BB,

^we^obtain

But we have from

[6]

1 For the sake of brevity ^thesymbol

has been employed ^to^denote

(4)

Hence

substituting

^from

(2.3)

ⁱⁿ^the

integrand

^onthe R.H.S. of

(2.2),

^and

changing

^the^order

of integration

^we

get

^the^result

(2.1 ).

To

justify

^the

change

of order of

integration

^{we see}^{that the}

t-integral

ⁱⁿ

(2.3)

^is

absolutely

^and

uniformly convergent

^{as n}

and s are

positive integers,

the

y-integral

ⁱⁿ

(2.1)

^is

absolutely

^and

uniformly convergent

and the

repeated integral

^is

absolutely convergent

^as^the

Meijer- Laplace

transform of

ItA ~0F(t, y)dyl

^exists.^Hence

by

^{De La}

Vallee Poussin’s Theorem

[7,

p.

504]

^theinversion of the order of

integration

^is

justified.

3. Particular cases

In all the

following

^cases^weassume n, s ^as

positive integers

with s

> n, arg ys/pn| (s-n)03C0/2

^{and the}

integrals

^involved

are

absolutely

^and

uniformly convergent.

(i) With aj

⁼

0, y =

^{1, 2,}^...,

m-1; am

⁼

-m’-k; 0’ = -m’-k, nm

⁼^m’-k^we

^have;

If

then

where

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4

min

for

Further with If

then

where

min

(Re ~j,

^Re

03C1)+

^Re

^03BB+1

> 0, ^Re

(~j+03B1j)

> -1

for j = 1, 2, ..., 03BC; l = 0, 1,

^...,^{n -1; v}⁼^{0, 1,}^...,^{s -1.}

(ib)

Further with k ⁼

±m’ and 03B1j

⁼

0, i

^-^{1, 2,}^...,^IÀ;^{p = 0}

we obtain If

then

With n = s ⁼1 we

get

^thewell-known Tricomi’s ^theorem

[5,

p.

564].

(ii) Setting 03B1j

⁼

0, j

⁼^{1, 2,...,}

^{m-1; am}

⁼

1 2-k-m’; nm

⁼

2m’,

^{a =}⁰^we^have

If

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then

where

for j = 1, 2, ..., 03BC; l = 0, 1, ..., 03C0=1; v = 0, 1, ..., s-1.

With 03B1j = 0, j = 1, 2, ..., 03BC-1; 03B103BC = 1 2-03BC’-03BB’; ~03BC = 203BC’; 03C1 = 0

and n = s ⁼1 we

get

^after^some

simplification

^the^result

by

^R.

Narain

[8,

p.

316].

EXAMPLE: The function

K(z)

of Boersma J.

[9]

^is^defined

by

where _{03BCj, vk}are natural

numbers; i

⁼

0, 1,

^...,

03BCj-1; j = ^{1, 2,} ..., y; l

⁼0, 1, ^...,

vk-1;

^k⁼1, 2,..., t ^and^const.

Let

in case

(ia)

^{of the}^theorem.

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6

Then

(3.1) gives,

where

F(t, y)

^is

given by (3.2), provided

^{n, s}^are

positive integers,

for j = 1, 2, ...,03BC; l = 0, 1, ..., n -1; v = 0, 1, ..., s-1;

03BCr, vt

are natural

numbers, 03A303BCj ~ 1+03A3vk; s

> n.

1 am thankful to Dr. B. R. Bhonsle for his

help

in the prepara- tion of this paper. 1 ^amalso thankful to the referee for his valuable

suggestions

ⁱⁿ

improving

the paper.

REFERENCES V. M. BHISE

[1] Jourl. Vikram Univ. 3, No. 3, (1959), p. ^57201463.

[2] ^Proc.Nat. Acad. Sci., India, ^32A(1962), p. 3892014404.

[6] Communicated for publication.

J. BOERSMA

[9] Compositio Math., ¹⁵(1961) p. 34201463.

BROMWICH T. J. I. ’A.

[7] Introduction to the theory of Infinite Series, ^London(1931).

C. S. MEIJER

[3] Proc. Ned. Acad. V. Weten. Amsterdam, ⁴⁴(1941), p. 7272014737.

R. NARAIN

[8] Rendiconti del Seminario Matematico dell’Universita e del Politecnico di Torino, ¹⁵(1955201456), p. 3112014328.

F. TRICOMI

[5] Rendic. Acc. dei Lincei, ²²(1935), p. 5642014571.

R. S. VERMA

[4] Proc. Nat. Acad. Sci, India, ^20A(1951), p. 2092014216.

(Oblatum 20-5-66) Department ^ofMathematics, G. S. Technological Institute,

IndoTe (India).

Certain properties of Meijer-Laplace transform

C OMPOSITIO M ATHEMATICA

V. M. B HISE