ON SOME INFINITE SERIES INVOLVING APPELL-POLYNOMIALS AND THE FUNCTIONS F (z,a)--i

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1 ON SOME INFINITE SERIES INVOLVING APPELL-POLYNOMIALS AND THE FUNCTIONS F (z,a)--i BY S. K. RANGAR/%JAN (Central Eleetroehemical Researeh Institute, Karatkudi, S. India) Received April 8, 1964 (Communicated by Dr. K. S. G. Doss, v.a.sc.) THE identity proved in tkis note is ~r (x) F (y, a q- r) zr = ~br ~ F (y -k- zx, a q- r) 3" (1) Herein, r! is an Appell-polynomial defined by r F (z, a) is the generating function (cfi TruesdelP) tional relation which obeys the func- F(z, a) = F(z, a + 1). bz The domain of uniform eonvergenee of the intinite series given by (1) depends upon the behaviour of the funetions ~b r (x) and F fz, a) chosen. The importante of this result arises from the faet that it gives a unified treatment to a large number of known summations besides suggesting many more. Among the elasses of funr fil (x) and F (z, a), we may mention the fouowing: 153

2 154 S.K.R.~GAR~Aaq ~, (x) (i) Bt. (x) r! xr Lra (k) (ii) 0~ (iii) x~'l~,fq [-- r' (a)p 9 ~] 605q' X (ir) (x' -F I)'/' P, (%/~) r! 1)~12 (v) (I + 2a)r (x2 + ST (-~)" (1.-t- a)r (- k)" (~,0,.... (~),. ~%+, = 0 ~o = ~_ 1>, ~~~: ~% = 0 (89 ~o (_ ~), (1 + ~), ~02t,+1 = 0 ~or = ( 1)1" (2r) l r! F(z, ~) ( F (a -- b + 1) (z ~ + 1)-(,+1)/~ p b ' ~ C(fi) e-" H, (-- z) (iv) z -"/~.l_. (2V'z). A r237 list is given in Referenee (1). The identity (1) can be provr by showing that the eoei of z m (m, any integer) on both the sides ate equal. * A misprlnt in ReŸ (1) is r h91

3 $inee On Some Infinite Series lnvolving Appell-Polynomials 155 z~ y" F(z-by, a) = F (z, a -b n) O (cf. Truesdel, x p. 82). We obtain for the coet r ~ F (z + tx, ao + r) tr of t ra in the intinite series to be N ~-~~. (m --r),f (z, a + m) whieh is the same as in the left-hand side of (1). We may rewrite (1) in the forro 'T ffr* (x) F (tan 0, + r) C sin o-~ V/r,. / (x sin 0 ---'~~ r/r. = ~br ~ F (tan ~b, a -b r) \cos O cos 41 [ I (2) where ~br* (x) is defined as A detailed list of sums of infinite series that can be evaluated from (1) has been drawn by the author, but for the sake of brevity, we present below only some of these as illustrations: 'I (a), (b)r -(c)r ~r(x)f(a+r,b+r; t r c+r,z)~ a)r Cb)r Oro F (a -t- r, b -k r; c -b r; z d- tx) ~ (3) = (c)r

4 156 S. K. R~G,tRiO,~',~ (a)r(b)r~br,(x)f(a+r,b+r; c+r;y) r! (c),. (_ y)r = (a)r (b)r ~bro (-- xy) ~" (4) (c)r r! (a)r t r (~)t ~r (x) qb (c -- a, c + r ; z) ~. = ext (a)r t r (-~)r ~br~ qb (c -- a, c -q- r; z -- tx) ~. (5) tl - cm,,. (x) 9 (a, c - r; z) -,l-,~)~ ~ (, ~,~~~~ (~So [o ~ ~ ~ ~,_t~~]/r, t6) (a)r.t, *(x)r a,c + r; y)~. = eu (a)r xry r (-~)t ~b'~ rt (7) ~b, (x) J,+r 2V/y (~y)r/r' _( y ~,/2 _~_)] -- ['Y~] ~r~ Ja+r [2"V"~ -'i- (,Vlfi-~t-tx))Tr' (8) Y-a/* ~br* I'x) Yr'*J"+r (2"V/Y)-~" -- ffr~ (~)r Some Particular Cases: --a result due to F(A,B; C; z) =(_l)r(a)1.(b)~. mo (c)rr! tfa (A, B, c, -- r ; 1 ) a, b, C; " z r 1 Pi1 +~+r)" (9),Fl(a + r,b + r; z~ c+r;, 0o) Chaundy--is a particular case of (3) (cf. Erdelyi,* p. 187).

5 It On Some Infinite Series Involving Appell-Polynomials can be easfly generalised as, [- r, (~v), c; x] ~"! 89 (~~r "+2Fq+~ I. (flq),a,b; r~fa(a91 c+r;z) F(a~);xz] (p~< +1;) = +,F~ L(&), q ni,~,(~ + 1),+ ~" tx) L~: t.v) z" {-- z(y-k X)} (l _ z)-, +x, -----exp 1 --z { xz. xyz q~a a -- [3, a,q- l, l _ z, (l ~ z~ J n! I.,n" z ~ (~ + 1). (x) ~~-'+ (y) ( = ) = exp (-- yz) (1 q- z)t3 (Pa -- q a q- 1, 1 q_ z ; xyz 157 (11) (12) (13) where ~, (t~, ~, x; y) = ~ (Dm xray r'. (y)rn+. m! n!' m a II (a)r (b)r Lr" (x) F (a q- r, b q- r; c -k r; y) (-- y)r (c)r (1 -k a)r = ~F~ (a, b; c; a + 1; xy) In particular, for b = 1 -k a, I.n. a (x) F (a + r, aq- 1 q-r; cq-r; --y) yr (a)r -- (c),. O = ~(a, c; - yx) (14) (15) (c+m--m 1)2 I'r" (x)

6 158 S.K. RANGA~AN F(--m+r,a+l +r; c+r;y)(--y) r(-m)r = Lm r (xy); (16) la+r (2V'y) l.,r a (x)yrli, (1 + a + r) y~/2 =F(l+a) P(i +q176 +a, 1 +/3; --xy) = Ja, ~ (3~/xy) (xy)-(a+a)/~ ya/~ (17) where Jm, n (x)is defined as, (~)~~ ( x.) Jn,,n(X)=F(m+l)P(n+l)oF2 m+l,n+l; " (18) In particular, ii"/3 = 0, we have, Lr a (x) y~'/2 q (1 J~. + (2V'y) a + r) = Ja, a ( 3 {/ x.3') "(x (y) ~-a/8 (19) --a result obtained by Varma (cf VarmaS). Also, tl,,,*o '~(-- 1)~wV,z)-(a+tt)~~ T. yf. ~, -a+v(2cz) oo = ~~I~~~~.+~.~+.~2~~,20) ~-0 obtained by TruesdeU (cf. Truesdell, 1 p. 134) follows immediately from (8) by assuming ~r (x) ---- (x + k) ~" and rearranging. In conclusion, the author thanks Prof. K. S. G. Doss, Director, Central Electrochemical Research Institute, Karaikudi, for Iris kind encouragement. 1. Tmegldl, C. 2. Erdelyt, A., et d. 3. Varma, R. S. RE~NCES..,4n F~aay Toward a UnO~d Tl~ory of 8p~kd Funetions Pfincr Higl~r Tran~cendr Function91 McGraw.Hill Book Co., New York, 1953, 1. _ Proc. lnd. Acad. Sal , 12 A, 532,