An extension of Neumann’s integralrelation for generalized Legendre functions

(1)

AN EXTENSION

OF

NEUMANN’S

INTEGRALRELATION

FOR

GENERALIZED

LEGENDRE

FUNCTIONS

BY IX. S. V.

DE

SNOO

In

this paper we obtain anintegralrelation connecting the

two

linearly

inde-pendent generalized Legendre functions of Kuipers and Meulenbeld. The

result is a generalization of

F. Neumann’s

relation of 1848 for the two kinds

of Legendrefunctions

1

_f

P(x)

@,(z)=’

j_

dx

where ]c_is _a _{nonnegative integer, and z is} _not_lying _on_the

_segment

_{(-1, 1)}

_of

the complex plane.

The main result is in

2;

generalizations can be found in

_4.

E. It. Love’s

integralrelations of 1965 for associated Legendre functions follow as special

cases.

1. The generalized Legendre functions

P’:’"(z)

and

Q’:"(z),

two specified

linearlyindependent solutionsof thedifferential equation

dw

{

m

n2

t

(1

--z

_2)-

_2Zz

-[- ]c(kq-

_{1)--2(1_z)}

_--2(1q-z)

w---- 0, have been introduced by Kuipers and Meulenbeld

[3]

as functions of z for all

points of thez-plane, in which a cross-cutexistsalong the real x-axisfromi to --o, and for complex values of the parameters

k,

m and n.

On

the

segment

--1 x

<:

1 of the cross-cut these functions are defined in

[7].

If m n,

they reduce tothe associatedLegendre functions, defined in

[2].

For

the sake ofbrevityweput

a /c+

1/2(m+n),

/ /c--

1/2(m--n),

lc

+

1/2(m--n),

lc

1/2(m

Generalized Legendre functions can be written in terms of hypergeometrie functions, such as

[4,

(9)]

(1)

Q’"(z)

e’’’2

F(a

+

1)F(,

+

1)

r(2/+2)

(z+

(z-+

,

+

;k

+

;

₊

if is not lyingonthecross-cut.

Received March 7, 1968. The author wishes to express his indebtedness to Professors

R. L.van deWetering andB.Meulenbeld. 71

(2)

72 H. S. V.DESNO0

2. The extension of

Neumann’s

integraliscontained inthefollowingtheorem. THEOREM 1.

T]

Re

m

,

1,

Re

>

-1,

Re

,

_>

-1, and z is not lying on the cross-cut, then

(z

-{-

1)*/

_{e_}

,Q

1

(1

+

x)*+

P[’(x)

(2)

(z--

1)

’(z)

(1 :x)

z--x dx.

Proo].

In

[8,

(13)]

one canfind the integral transform

(3)

(1

x)-t(1

+

x)"(z

+

x)-P2’(x)dx

2.+t._+,

r(q

+

n

+

1)r(q

n

+

1)

r(q

m

+

)r(q

+

m

+

2)

(z- 1)

aF(qWn+

1, a,q--

n+

1;

if

Re

m

<

1,

Re

q

+

1

>

IRe

hi,

z not lyingonthecross-cut. Choose a 1, q k

+

m

and replacezby -zin

(3);

then

+

x)

+t

P

(4)

_a-(1

x)

-

z-- x

’’(x)

dx

r(2+2)

(z+l)

+

2+2

if

Re

m

<

1,

Re

a

>

-1,

Re

y

>

-1, znot lyingonthecross-cut.

Using

(1)

and therelation

[5,

(6)]

(5)

Q;"-"(z)

e-:"2

F(

+

1)F(

+

1)

r(

+

)r(

+

)Qr’"(z)

weobtain

()

F

_+’+;k+;i+

e

’2

-r(

+

)(

+

)

(z

+

)

."(z).

By

setting

(6)

in

(4)

wecomplete the proofof the theorem.

emar

1. The above theorem is a generalization of results of Meulenbeld

and Robin

[9,

(30)

and

(31)].

After some simplifications

eumann’s

integral follows fromTheorem 1 form n 0 and nonnegtive integer

.

(3)

GENERALIZED LEGENDRt FUNCTIONS 73

LEMMA

1.

then

(7)

(8)

/1 Re

m

_<

1 and is anintegersatisfying

0

_

_i

_<

min(Re

1

+

x)

(1 --x)

im

pm,

n(x)

dx

O. In

[1, (8.1)]

we find the integraltransform

(1

x)’(1

+

x)’P’’"(x)dx

r(q

+

1/2n

+

1)F(p

1/2m

-b 1)

r(1

m)r(p

+

q

1/2(m -n)

-

2)

F(fl-{-

1,--7,p--

1/2m-

1;1-- m,p-[- q--

1/2(m

n)

-b

2;1)

!_m

if

Re(p- 1/2m)

>

1,

Req+

1

_>

_1/21Ren

I.

Substitutep

-

,q

k+

1/2m

iin

(8);

then

(9)

(1

x)-tm(1

₊

w w

,F(a--i)

_F(+

_1,--;--i+

_1;1)

2a-’

r(

₊

)

if

Re

m

<

1,

Re

_<

min

(Re

a,

Re

7).

Using

[2,

2.8

(46)]

therighhand side of

(9)

reduces to

a-i

r(- i)r(=

)

r(-)r(-

+

2

+

1)

which isequal to0forj satisfying theconditions of

Lemma

1.

LEMM

2.

I] Re

m

_<

1,j is any integersatis]ying

0 j

_<

min(Rea,

Re)

and

p(x)

is any polynomial

o]

degree orless, then

]’(1

+

x)

+--Pro4.

Leg

_p(z)

beapolynomial of degree orless. hen

p(z)

canbewfiggen

inheform

(z

p(

+

z

-’.

This yields

(1

q-

x)

(1

x)

f_’

(

+x)/

p(x)P"2’n(X)

dx i-oPi

(1

x)

i

By

virtue of

Lemma

1 the right-handside vanishes.

(4)

74

.

s.v.D.SNOO

4. This section contains two generalizations of Theorem 1.

presented aresimilartothose in

[6].

The proofs

THEOREM

2.

I] Re

m

_<

1,

Re

c

>

--1,

Re

3"

>

-1, z is not lyingon the

cross-cut,

sisanintegersatisfying

0_< s

_<

min(RecA- 1,

Re3"A-1)

and

p(x)

isa polynomial

_of

degrees orless, then

(11)

(z 4- 1)

+m- 1

(1

-b

x)

+-

P

....

(z-

1)

Ira-

e-’"Qr"(z)P(Z)

(1- x)

1/2m _z-x(X)

p(x)

dx.

Proof.

Using Theorem 1 and subtracting the right-hand side from the left-hand side in

(11)

we obtain

=0

because for 1

_<

s

<

min

(Re

a

_-t-

1,

Re

3’

-t-

1)

the expression between braces

isapolynomial inxof degrees 1, sothatwe canapply

Lemma

2by putting

i

s 1.

For

s 0 the polynomial p(x) reduces to a

constant,

and

(11)

follows fromTheorem 1.

Remar

2. The above theorem is a generalization of results of M:eulenbeld andRobin

[9,

(53)

and

(55)].

THEOREM

3.

I] Re

m

_>

--1,

Re/

>

--1,

Re

_>

--1, a isnot an integer, z is not lyingonthe cross-cut,sis aninteger satis]ying

0_< s

<

min(Re/-t- 1,

Re

_A-

1)

and

p(x)

is any polynomial

o]

degreesorless, then

(12)

_L(z

_{1)_,}

e

Q (z)p(z)

i

{s

f_

(i

+

x)-l-"

P"(x)

2sinai-

(1--

_Xi

’-t z--x

(13)

p(x)

dx sinmr

F(3" A-1)

L

(1

x)-t-,

p.m.()x

x}

F(BA-

1)

(1-x)

-:"

xP(-x)

d

By

combining

[7,

(8)]

and

[7,

(10)]

we obtain, if a is not integer

r(

+

)r(

+

)

r(

+

1) r(

+

1)

Pi"’-(x)

1

₍₂

sin

flrP’’(x)

+

sinmr sina

F@

-b 1)

_p.t

)

r(+ 1)

t-z)

(5)

GENERALIZED LEGENDRE FUN CTIONS 75

In

Theorem 2 replace m by -m and n by -n, use

(5)

for the left-handside

and

(13)

for the right-hand side, split up the integral in the right-hand side

(this is allowed because both integrals exist under given conditions) and in

the integralinvolving

P’(-x)

change x into -x. This completes theproof. Remark 3. All results given in this paperreduce to

E. R.

Love’s

results

[6]

by setting m n.

For

references to special cases in the case of associated Legendre functions,we referto

[6].

REFERENCES

1. B. L. J. BRAAKSMA AND B. MEULENBELD, Integral transforms with generalized Legendr

functionsaskernels, CompositioMath.,vol. 18(1967),pp.235-287.

2. A. ERDELYI, Higher Transcendental Functions I, NewYork-Toronto-London, 1953. 3. L. KUIPERSAND B. MEULENBELD, Ona generalizationofLegendre’s associateddfferential

equation, I and II, Proc. Kon. Ned. Akad. Wet., Amsterdam, vol. 60(1957), pp. 436-450.

4. L. KUIPERS AND B. MEULENBELD, Linear _{transformations} _of generalized Legendrv’8

associated functions, Proc. Kon. Ned. Akad. Wet., Amsterdam, vol. 61(1958),

pp. 330-333.

5. L. KUIPERS AND B. MEULENBELD, Some properties _ofa class of generalized Legendre’8

associated functions, Proc. Kon. Ned. Akad. Wet., Amsterdam, vol. 61(1958),

pp. 186-197.

6. E. R. LOVE,FranzNeumann’sintegral_of1848,Proc.Cambridge Philos.Soc.,vol.61(1965),

pp. 445-456.

7o B. MEULENBELD, GeneralizedLegendre’sassociatedfunctionsforrealvaluesoftheargument

numerically less han unity,Proc. Kon. Ned.Akad. Wet.,Amsterdam, vol. 61(1958),

pp. 557-563.

8. B. MEULENBELDAND H. S.V.DE SNOO, Integralsinvolvinggeneralized Legendrefunctions,

J. Engrg. Math.,vol. 1(1967),pp. 285-291.

9o B. MEULENBELD AND L. ROBIN, Nouveaux rsultats relatifs aux _fonctins de Legendr

gnralises,Proc. Kon.Ned. Akad.Wet.,Amsterdam,vol.64(1961),pp. 333-347.

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