A N O T E ON T H E G E N E R A L I S E D L A G U E R R E P O L Y N O M I A L

BY HAm DAs BAGCm (Calcutta Uniwer, ity)

AND BHOLA NATH MUKHERIEIE

(S. C. College, Calcutta)

Received November 19, 1951 (Conununicated by Prof. B. S. Madhava Rao, Iv.A.-~--)

1. The gencraliscd Laguerre polynomial

e ~ z--" ( e - : . z ~ ) , L,, '~ (z) = -~i!-

where a > -- I and n is an integer ~> 0, is a solution o f the fo l lowing differen- tial and functional equat ions :

_,t-f ,,(z) a ~ : z ) , , l . . (z) o., - -d i i V f f , - l - - ) - - - - + =

(1)

, f , , (z) - - (2n - - a - - 1 - - z ) f , ~ x ( z ) + (n + a - - 1)]' ._,(2:) = 0 (2)

f ' , , ( : ) ----- f',,--1 (z) --f,,--1 (z) O ) (,t e. - - z ) f ' , , ( z ) = (n ÷ a ) f ' , , - I ( z ) - - . f . , ( z ) (4) z f ' , ( z ) -= n f ,~(z) - - (n + a ) f ,~_ 1 (z) . (5)

It is easy to ~erify that these are equivalent to t ~ o independent equat ions only.

Consider now the series

v = s h 1,, (_), (6) 0

~ e r ¢ f , , (z) is an analytic solution el" (2). When [ h [ < R. the radius o f convergence o f the power series (6),

bV "= - = z , O : - l f , , ( z ) . (7) ~h .= .

Multiplying (2) by h ~1 and summing up for n = 2, 3, 4 . . . . . . we get O 0 0 0

X n h " - x f , , ( z ) = 2 X n i t ' L - l f , , . . x ( Z ) + (a - - 1 - - z ) Z ' h " ~ l f r ~ ( z ) ~ s = ~ _ l # "P • •

0 0

- - X(n - - 1) h'~-l f ,,_.. (z) - - a ,V ,h" - l f , r_~ ( z ) . s ~ ' t t n ,

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54 ItARI DAS BAGCItl AND BilOLA NAIl! MUKltEILIEE

which can bc written, by using(6) and (TL as

~ V , ( a - i - l ) h - - A 3h ' (! " -h ) ~ V =

where A ~ a + I -- c and B = f x ( : ) - - A fu (z )-

Solving (8),. we get

V ~ . . . . . . . it . . . .

(1 - - hi"-' e i=¥

B ( I - ~ h)"' (8)

- ( I h l < R ) . ( 9 )

To determine d, (z), take the particular x~!ue h ----- 0. Then (9) gives

(z) = g f o (z).

The value of the radius o f ~xmxergencc R is obtained by applying Van Vleck's theorem ~ on Linear Cri~eEa for the dci,:rminatmn of the radius of com'crgence of a power series. From !2).

f ,,+.,_(z) = & f .._, (_-~ + q , L (:) 2 ~ | I ~ - a + 3 - - z n-r- ! + ~ t

where p , = and q,, - - n + 2 n + 2

Since p=--+2, q, , - - - - - - I (when n - - , . ~ ) . R is equal to the Coincident root o f the quadratk k - ° - 2 k - I = 0. i.e.. R = 1. V¢c thus have:

I f f , , (z) be aJo" anal)'tic sohaion , , f the d(~lbre,:ce eqttati~,n

(n + 1Lr,,_~ ( z ) - ( ~ , - ,~ - I - : ) f ~ , ~2 ) - ' ( , - ~ ) f , , . . . , ( z ) = 0

attd the pobtt " h" is hlside the unit cir~h" :~-#h centre at the origbt, then

{A (z) -- (a -T i -- :).1;(:)1 i -- t) ":~ o - , ,It + e:fo (z) ~ h " f , ,(z) =

.=0 ( I - - h ) *+t e ~ - i (10)

the lbw-bltegra! beblg taken along any Jordan cu~'e coanecling 0 and" h "

3. I f we set f o ( z ) = Lo~*~(z)= i . . A ( z ) = L l ~ = ~ ( z ) - - a + ! - - z , the formula (10) leads to the result s

~ ' : . c , ," ( . .O = (l - h V ' - " , ' - , - ~ \ , ( l h l < l). Jt ,~-.@

(10

A Note cm the Gcncralised Lagtwrrc Polynomial

If F (:) :.dmils of ~: uniformly convcrocnt expansion of the typc o o

F ( 0 = ZA,, L.'"~ (z ) , P l = ( )

it is kno~vn that -~

Taking

o o

I t ! ¢" .

A,, I '(a + ,~ + !) .,I . F (x) L,, '"~ (x) dx. II

- - J i b

F(z) ::- , ' , -~ , we obtain 4 from (l l), o o

f ,,-1"-, x" L,,t')(x)d:r = o

[ ' (n . ._~ ' o . ~ I I . __'_ -h" d _ _ h)~t n !

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REFERENCES

1. Van Vleck Trans. Amcr. ,~fath. Sac., Vo]. I , 1900. 2. Th!~ result was obtained by V. R. Thiru~enkatachar b', a different method, Proe.

Ind. Acad. Sei. , A, 1940, 229-34. 3. Copson . . Tb.eorv ¢~f Ftmctio~rs o f a Complex Variable, p. 269, Ex. 22. a. This ,,,,-as obtained by V. R. Thiruvenkatachar (loc. cir.), by a different method.