GENERATING FUNCTIONS FOR JACOBI AND RELATED POLYNOMIALS I. A. KHAN

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 1, March 1972 GENERATING FUNCTIONS FOR JACOBI AND RELATED POLYNOMIALS I. A. KHAN Abstract. In the present paper, we have established the following relation involving Kampe de Feriet's double hypergeometric function of superior order «i, ' ' ',"p Ci. b, -b', ca -xy, -y = y ia_(-l)"(-6') -n,b; 3=1.1 +*'-»; which yields a number of interesting generating formulae for Jacobi ( and related polynomials. A large number of special cases have been also discussed. 1. Introduction. In this paper, we establish some generating relations of general nature for Jacobi, Gegenbauer and Legendre polynomials, employing an arbitrary sequence of parameters. Certain particular cases yield several interesting generating relations for Laguerre polynomials also. The results proved here lead to certain generalizations of the various well known formulae due to Brafman [2], Carlitz [3], Feldheim [6], Jain [7], [8], Manocha [9], [10] and several other results also. The Kampe de Feriet's double hypergeometric function of superior order [1, p. 150] is defined by (1.1) P ay,, ap by, by, ; b, b' cl> ', cv dy, d'y\- ; ds, d's x,y = I m,n=0 IT (aj)m+n IT {(b^ráb'j)»} m n 3=1_3=1_ * y ml In' n IT (Cj)m+n IT [\d])m(d',)n} 3=1 3 = 1 W<L M<1. Received by the editors January 22, AMS 1970 subject classifications. Primary 33A65, 33A30, 33A45; Secondary 42A (c) American Mathematical Society 1972

2 180 I. a. khan [March The Jacobi polynomials [11, p. 254 (1)] are defined by (u) iew-^r*a[?,;t"*'+",1ri n' L 1 + a; 2. The Gegenbauer polynomials [11, p. 280 (19)] are defined by (1.3) n n + 1, (fe)n(2x)" C (x) =-;-2Fj n! fr-n; which reduces to Legendre polynomials Pn(x) [11, p. 166 (4)] on putting b=h The Laguerre polynomials [11, p. 200 (1)] are defined by 1 (1.4) Un (x) =-;-ifji-n; 1 + a; x). 2. We require the following formulae to prove our results: (2.1) 2Fx[a, b; 1 + a b; z] = (1 + z)-^ which is formula (2) in [11, p. 66]. (2.2) a a ~a, b; 4x «Fi (i + x2ra(i + x)2a2f1 2b; (1 +x)2_ which is a formula on p. 66 in [11]. ;\ + a- b;az(\ + z)~ a a + 1 2' 2 b + è; 4x2 (1 + x2)2 (2.3) Fx(a, ß, ß', ß + 0', x, y) = (1 - TV^a, /S;/3 + ß'; *-=^l where Fx is Appell's first function [4, p. 224]. This is formula (1) in [4, p. 238]. In (2.3), replacing x by x/a, y by j/a and making a-*-oo, we get (2.4) 02(p\ ß\ ß + ß>, x, y) = e\f^; ß + ß';X-y), where 02 is Horn's function [4, p. 225, (21)]. O c\ pu n. a y n)/\ _ / 1 + X1 p(a n.v-1) /-» XV (2.5) Fn W \ 2 r * ll+x/' which is a relation in [12, p. 63].

3 1972] GENERATING FUNCTIONS FOR JACOBI POLYNOMIALS Let us consider (h\ (~b') Cm+n- m.n=o mini (-xy) (-y) (3.1a) co n (b) ( h'\ 2 C«~~Tr-ÏÏ (~y) X =o m=o m\(n m)\ n=0 n\ n, b; Ll + A'- n; y", where {Cn} is an arbitrary sequence (subject to convergence conditions). In (3.1a), replacing the arbitrary sequence {Cm+n} by 11 \aj)m+n n>,)..3=1.3=1 (a, c arbitrary parameters), and making use of (1.1), we obtain the following relation : P 1 a LO ay,---,a b, -V Cy,, cg -xy, -y (3.1) = 2 - n, fc; Fy.1 + A'-»;"J 3=1 In (3.1) replacing x by (1 x)/2 and applying (1.2), we have a generating relation for Jacobi polynomials (3.2) P 1 a LO ai,, s b, -V Cl> ' ' ', Cq (x - l)y 2 -y oo IT K')n _ -^ 3=1_ 3=1 P{:'-n^-1)(x)yn, \(x - l)y/2\< 1, Lv <l. Replacing x by (3.x)/(l+x), y by (1 + x)y/2 and b by and making

4 182 I. A. KHAN [March use of (2.5), the above formula (3.2) assumes the following form: (3.3) fli, - - -, ap (1 - x)y -b, -V ^i> > <-ï -(1 + x)y oo Tiiai)n = 2^n=0 n(^) 3=1 ix)y, (1 - x)y <1, (1 + x)y < 1. Further, supposing u=t (t2 1)1/2, v=l/u=t+(t2 1)1/2, replacing x by m2, j by _y/w and b' by /», and making use of (2.1) and (1.3), the relation (3.1) yields the following generating relation for Gegenbauer polynomials (3.4) a3,, av b,b uy, vy oo tliai)n = 2i=Jn=0TlicX 3 = 1 Clit)y\ \uy\<l, \vy\<\, u = t-(t2-\)m, v=t+(t2-l)m. Putting b=\ in (3.4), we get a generating relation for Legendre polynomials: (3.5) ax,, a i i 2^ 2 ClJ ' " ' > C«t/j», uy oo IT Oí)«= 2if-/' (O/, 3 = 1 uy <l, t»y <l, u = t-(t2-\f'2, ü=í+(í2-1)1/2. Throughout all our results, u and v stand for the same values as given above. 4. Particular cases. I. In (3.2), puttingp=q=0, b' = a, b=l+a.+ß, we get a known result [6, p. 120] due to Feldheim. II. In (3.2), puttingp=q=\, b' = ct., b=l+a.+ß, we get (4.1) (x - 1)3» FJa,, 1 +a + ß, -a, cl5,-y)=l^p{r-ßxx)y\ i «-o (cl)n (x - 1)J» < 1, \y\ < 1.

5 1972] GENERATING FUNCTIONS FOR JACOBI POLYNOMIALS 183 Putting cx=\+ß and applying (2.3), the result (4.1) degenerates to (4.2) (i + y)~a\ y(i + x)\ ofyiay, 1 + a + ß;l+ß; 2(1 + y)) ^ (al)n (x-n.ß), s n «=0(1 + /S)n III. In (3.2) puttingp=0, q= 1, b' = ct, b= l+a+ß, we get (4.3) <D2il +a (x - \)y + ß, a, Cy, )00 1 Putting c1=l+/5 and applying (2.4), the result (4.3) reduces to 1 (4.4) e-\fy(i TaT^l+^;y(lT x)/2) = 2 P(rn-"\x)yn. n=0 (1 + ß)n IV. In (3.1), putting p=q Q, replacing x by x/b, making Z> >-oo, and using (1.4), we get (4.5) e-*»(i + yf = Z» W, \y\<u n=0 which is relation (19) in [5, p. 189]. V. In (3.1), puttingp=q=l, replacing x by x/è and makingb^co, and using (1.4), we get 00 (a ) (4.6) x(ax, -b', cy, -y, -xy) = 2 ~V lt""w\ M < 1- where <!>! is Horn's function [4, p. 225 (20)]. Putting cx= b', the result (4.6) reduces to 00 (n \ (4.7) 2~(-jOV1 d=o nl ay + n; -b' + n; -xy = 2 7 Ln (x)y. n=0( b )n VI. In (3.1) taking />=0, <7=1, replacing x by x/ > and making è >-oo, and using (1.4), we get (4.8) %(-b', cu -y, -xy) = 2 «=«(ci) lî^w, where 03 is Horn's function [4, p. 225 (22)]. Putting Cy = V in (4.8), we get (4-9) 2 -,ofi n=0 nl -V + n xy (b' - h)! Lv" = 2 1 6'! - (&' n)sv\..n (x)yn

6 184 I. A. KHAN [March VII. In (3.3), putting p=q=0, we get a well-known result [3, p. 88] due to Carlitz. VIII. In (3.3), putting p=q \, we get the Manocha's result [10, p. 457 (1.4)], which reduces to another result of his [10, p. 457 (1.5)] with the help of (2.3). IX. In (3.3), putting/»=0, q=\, we get (4.10) *J-b, -b', Cl, -4^, =fi±5iz) = i -V pf:-"^\x)f. \ 2 2 / n=o (cx)n In (4.10), putting cx= b b' and applying (2.4), we get (4.11) exp "-(1 + x)y xfx[-b;-b-b';y] = 2 T-r T^ C~",8~"W> B-o( b b')n which is result (1.4) in [9] due to Manocha. X. In (3.4), putting q=p, we get a known result [7, p. 28 (2)] due to Jain. XL In (3.5), putting q=p, we get a result [7, p. 29 (5)] due to Jain. XII. In (3.4), putting q=p=\, we get another Jain's result [7, p. 29 (6)]. XIII. In (3.5), putting q=p=\, we obtain a result [7, p. 29 (7)] due to Jain. XIV. In (3.4), taking/»=0, q=\, we have 00 1 (4.12) < > b, b, Cl, uy, i»y) = 2 rv C W>'B- XV. In (3.5), takingp=0, q=l, we obtain (4.13) 02(i, J, cls uy, t»y) = 2 tv Fn(0yn. XVI. In (3.4), taking p=q=0, we get a known result [11, p. 276 (1)]. XVII. In (3.5),taking/»=^=0, we get a known formula [11, p. 157(1)]. XVIII. In (3.4), putting/»=0, q=\, cx=2b and applying (2.4), we get (4.14) exp(t;y) xfx\b; 2b; (u - t»)y] = 2 - Cbn(t)yn, n=0 (2b)n which is a relation [8, p. 109 (2.1)] due to Jain.

7 1972] GENERATING FUNCTIONS FOR JACOBI POLYNOMIALS 185 On applying Kummer's second formula [11, p. 126 (9)] formula (4.14) assumes the form : (4.15) pyt F e ori ' _; y2(t2 - iy.b + h 4. = 2 i»-o (2b)n Cl(t)yn, which is a known formula [11, p. 278 (7)]. Putting i=cos 0 and using the definition of Bessel functions of first kind, the above result (4.15) changes to the well-known relation [5, p. 177 (30)]. XIX. In (3.5) puttingp=0, q=i, cx=\ and applying (2.4) we get (4.16) 00 P (t) exp(vy) yfy(%; 1 ; (u - v)y) = 2 T" / =o nl Making the use of Kummer's second formula, the result (4.16) yields the known relation [11, p. 165 (4)] which, on using the definition of Bessel functions of first kind, assumes the form of the formula [5, p. 182 (40)]. XX. In (3.4), putting q=p= 1, Cy=2b and applying (2.3), we obtain (4.17) (1 - vynfyiay, b; 2b; (~^) - C»( * \ 1 vy 1 n=o (2b)n Employing (2.2) in (4.17), we obtain a known result [11, p. 279 (8)], given below a.siij;^..!) 00 (a \ (4.18) (l-tyt%fy 2V"l/7r /^b/,\ n Cn(t)y. (1- tyf «-o (2i>) b + l, Changing the result (4.18) into the form of an ultraspherical polynomial, with the help of [11, p. 277 (5)], we obtain a well-known result (18) in [2] due to Brafman. XXI. In (3.5), putting q =p=l, cx=l and applying (2.3), we have (4.19) (1 - vy)^2fy(ay, *; 1 ; (^A = 2 ^ P»«>^ V 1 vy I n=o n! Using (2.2), the formula (4.19) yields the well-known formula (21) in [2] due to Brafman. I wish to express my sincere thanks to Dr. R. C. Varma for his kind supervision during the preparation of this paper. Further, I owe a very large debt of gratitude to the referee also, for his valuable suggestions to improve the paper.

8 186 i. a. khan References 1. P. Appell and J. Kampe de Feriet, Fonctions hypergeometriques et hyperspheriques, Gauthier-Villars, Paris, F. Brafman, Generating functions of Jacobi and related polynomials, Proc. Amer. Math. Soc. 2 (1951), MR 13, L. Carlitz, A bilinear generating function for the Jacobi polynomials, Boll. Un. Mat. Ital. (3) 18 (1963), MR 27 # A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill, New York, MR 15, , Higher transcendental functions. Vol. II, McGraw-Hill, New York, MR 15, E. Feldheim, Relations entre les polynômes de Jacobi, Laguerre et Hermite, Acta Math. 75 (1942), MR 7, R. N. Jain, A generating function for Gegenbauer polynomials, Ricerca (Napoli) (2) 20 (1969), gennaio-aprile, MR 40 # , Some generating functions for Gegenbauer polynomials, J. Reine Angew. Math. 235 (1969), MR 39 # H. L. Manocha, Bilinear and trilinear generating functions for Jacobi polynomials, Proc. Cambridge Philos. Soc. 64, (1968), MR 37 # , Some bilinear generating functions for Jacobi poylnomials, Proc. Cambridge Philos. Soc. 63 (1967), MR 35 # E. D. Rainville, Special functions, Macmillan, New York, MR 21 # G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc, Providence, R.I., Department of Mathematics, Government Engineering College, Rewa (M.P.), India