On higher differences. Nota II

(1)

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

S. C. C HAKRABARTI On higher differences. Nota II

Rendiconti del Seminario Matematico della Università di Padova, tome 23 (1954), p. 270-276

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(2)

Nota II

(*)

di S. C. CHAKRABARTI

( a Jadab pur College, Calcutta)

II. - The Relations between the Differential Coef f icients

and

the Higher Differences

of a

Function.

1. Introduction. This paper is in continuation of Note I

on the

subject

^of

Higher

Differences. We make use of the same

notations as

employed

in Note I. The

object

^of^thispaper is to

study

^therelations between the

operators

^{and the}

operators

ⁱⁿDifferential Calculus.

2. LEMMA

(i).

then

similarly

^formed

evidently develops

^into

.

Similarly develop Zn,

and then prove the theorem

by

^in-

duction].

(* j Pervenuta in Redazione il 26 novembre 1953.

(3)

271 LEMMA

~ii).

^If

then

[Here develop P fte

in t~rms of

C,

^C2^etc.^{and then}

apply (1)

above and Th.

( 5),

^Note

I].

LEMMA

(iii).

^-^If

(3)

then

3.

By

Differential Calculus and

by § II,

^Note

I,

where d"s

uo stands for the value Of

dm

u, when ^cis re-

placed by

^0.

Thus the

operators

^F^arerelated to the

operators

ⁱⁿ^Dif-

ferential Calculus.

(4)

4.

A nux

may be

expressed

ⁱⁿ^terms

of the differential coef f icients of

_ux..

THEOREM. If it, be ^arational and

integral

function of x

of

degree I in x,

^then

’

where stands for the value of when x - 0.

By

^Th.

(26),

^Note

I,

^have

In this

equation, putting n

⁼

1, 2,

^{3 and}

4,

^we^{have four}

equations

^from

^which eliminating Aux, A2ux

^and ^we

have

(5)

273

The

general

^case^{may be}

^similarly

^treated.

COR.

[This

result of Finite

Differences,

follows from

(5)

when

a ---L- 1,

^for

A~==A~=A~=... = ånon-l = 0

^and

AnOn = n!].

5. Anus

^may^{also be}

expressed’

ⁱⁿ^terms^of

^{the diffe-}

rential coefficients of

ux.

THEOREM. If Us ^be^arational and

integral

^function^of^a?

of

degree I

^then

°

where denotes what becomes when ^x⁼0.

(6)

If in this

equation

^we

put n = 1, 2y

^..., n, ^wehave n

equations

from which

eliminating

for = 0 when _{p n .}

This follows from

(6)

^{when a --}^1. ,

(7)

275

6.

dx

^may^be

^expressed

ⁱⁿ^terms

^oi ^{A 1,}

^’

^{A 3,}

^etc.

THEOREM. If Us be ^arational and

integral

^function^of^x

of

degree

ⁿin x, then

By

^Th.

(26),

Note I and

by (4), § 3,

^we^have

Let us consider the

particular

^case

when ux

^is^a^rational

and

integral

f unction of s of the third

degree.

^If^we

put

ⁿ⁼¹

in

( 9j,

^we^have

Putting

^{n =}^{2 and}

3,

^two^similar

equations

may be ob- tained. From _~ these three

equations, eliminating

^s’ ^g

tfu. _dX2

^and

we have

The

general

^casemay be

similarly

^treated

COR.

which is a well-known theorem in F. D.

(8)

1,

^thecoefficient of

(_)r-1A.Tux

ⁱⁿ

(8)

^becomes

which may be written

by

Finite Differences

Hence

(10)

^follows^from

On higher differences. Nota II

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

S. C. C HAKRABARTI On higher differences. Nota II

Rendiconti del Seminario Matematico della Università di Padova, tome 23 (1954), p. 270-276

(*)

( a Jadab pur College, Calcutta)

II. - The Relations between the Differential Coef f icients

the Higher Differences

Function.

subject

Higher

employed

object

study

operators

operators

(i).

similarly

evidently develops

Similarly develop Zn,

by

duction].

~ii).

[Here develop P fte

C,

apply (1)

( 5),

I].

(iii).

(3)

By

by § II,

I,

dm

placed by

operators

operators

A nux

expressed

of the differential coef f icients of

integral

degree I in x,

By

(26),

I,

equation, putting n

1, 2,

4,

equations

which eliminating Aux, A2ux

general

similarly

[This

Differences,

(5)

a ---L- 1,

A~==A~=A~=... = ånon-l = 0

AnOn = n!].

5. Anus

expressed’

the diffe-

rential coefficients of

integral

degree I

equation

put n = 1, 2y

equations

eliminating

(6)

dx

expressed

oi A 1,

A 3,

integral

degree

By

R ^ENDICONTI

S ^EMINARIO M ^ATEMATICO

U NIVERSITÀ DI P ^ADOVA

^which eliminating Aux, A2ux

^similarly

^{the diffe-}

^expressed

^oi ^{A 1,}

^{A 3,}

tfu. _dX2