Five-term recurrence relations for hypergeometric functions of the second order

Transcription

1 South Asian Journal of Mathematics 2013, Vol. 3 ( 2): ISSN RESEARCH ARTICLE Five-term recurrence relations for hypergeometric functions of the second order A. Kumar Pandey 1, R. Bhardwaj 2, K. Wadhwa 3, Nitesh S. Thakur 4 1 Department of Mathematics, Patel Institute of Technology, Bhopal, M.P., India 2 Department of Mathematics, Truba Institute of Technology, Bhopal, M.P., India 3 Department of Mathematics, Govt. Narmda P.G. College, Hoshangabad, M.P., India 4 Department of Mathematics, Patel College of Science & Technology, Bhopal, M.P., India pandey1172@gmail.com Received: ; Accepted: *Corresponding author Abstract Five-term recurrence relations for hypergeometric functions of the second order are deduced from generating functions involving elementary functions. Generalizations is also given. Key Words fixed point, weak contraction MSC H10, 47H06 1 Introduction and Preliminaries It has recently been pointed out by Y?a?nez, Dehesa and Zarzo [8] that recurrence relations for hypergeometric functions of the second order are incompletely known except for certain three-term recurrence relations associated with orthogonality properties. This is all the more remarkable in view of the fact that these functions are of importance in many applications including mathematical physics. The authors cited above have discussed certain four-term recurrence relations for these functions from the point of view of their associated differential equations. In this study, this matter is taken up beginning with certain generating functions which yield higher-order functions from which the required recurrences are deduced.if the higher-order functions are suitably selected, then they may be specialized to give the second-order functions concerned, and, in turn, the recurrences sought. In the present paper we find five-term recurrence relations for these functions from the point of view of their associated differential equations, which is motivated by Yanez R. J., Dehesa J. S. and Zarzo A.[8] and Exton Harold[2]. The following functions occur in the subsequent analysis: The hypergeometric function of general order, AF B (a 1, a 2,, a A, b 1, b 2,, b B ; x) = A F B ((a); )(b); x) (1.1) Citation: A. Kumar Pandey, R. Bhardwaj, K. Wadhwa, Nitesh S. Thakur, Five-term recurrence relations for hypergeometric functions of the second order, South Asian J Math, 2013, 3(2),

2 A. Kumar Pandey: Five-term recurrence relations for hypergeometric functions of the second order The kempé de fériet function F (A:B;B ;) (C:D;D (a):(b);(b ;) [ );) (c):(d);(d ); x, y] = ((a), m + n)((b), m)((b ), n)x m y n ((c), m + n)(d, m)((d ), n)m!n! (1.2) The humbert function And the first Appell function Φ 2 (a, b; c; x, y) = ((a, m)(b, n)x m y n ) (c, m + n)m!n! (1.3) F 1 (a, b, b ; c; x, y) = ((a, m + n)(b, m)(b, n)x m y n ) ((c, m + n)m!n!) (1.4) The pochhemmer symbol is (a)n = a, (a + 1),, (a + n 1) = (Γ(a + n))/γa; (a)0 = 1 (1.5) 2 Five-term recurrence relation for Gauss function 2 F 1 We consider a generating function U = (1 xt) a (1 yt) b (1 zt) c (1 wt) d = t n F n (2.1) The product of four binomial function is developed in power of t (1 xt) a (1 yt) b (1 zt) c (1 wt) d = ((a, p)(b, q)(c, r)(d, s)) p!q!r!s!x p y q z r w s t p+q+r+s (2.2) Taking p = n q r s, we get Within its domain of absolute convergent, we get t n [ n! (a, n q r s)(b, q)(c, r)(d, s) p!q!r!s!x n q r s y q z r w s t n (2.3) x n ( n, q + r + s)(b, q)(c, r)(d, s) ( y (1 a n, q + +s)q!r! x )q ( z x )r ] (2.4) By comparison with generating function and using appell series it is clear that F n = F n (x, y, z) = F n (a, b, c; x, y, z) = x n F n ( n, b, c, d, (1 a n); y n! x, z x ) (2.5) The number of functional relation may be obtained by generating function by taking partial derivatives with respect to t. Differentiate (2.1)partially with respect to t, we get U x = ax(1 xt) a 1 (1 yt) b (1 zt) c (1 wt) d + by(1 xt) a (1 yt) b 1 (1 zt) c (1 wt) d +cz(1 xt) a (1 yt) b (1 zt) c 1 (1 wt) d +wd(1 xt) a (1 yt) b (1 zt) c (1 wt) d 1 = nt n 1 F n (2.7) 110

3 South Asian J. Math. Vol. 3 No. 2 Rearranging, we have = [ax(1 yt)(1 zt)(1 wt)+by(1 xt)(1 zt)(1 wt)+cz(1 yt)(1 xt)(1 wt)+dw(1 yt)(1 zt)(1 xt)] t n F n = (1 yt)(1 zt)(1 wt)(1 xt) nt n 1 F n (2.8) Equating coefficient of tn and obtained the five- term recurrence relation (n+1)f n+1 = F n [ax+by+cz+dw+n(x+y+z+w)]+f n 1 [xy(a+b n+1)+yz(b+c n+1)+xz(a+c n+1) +xw(a + d n + 1) + yw(b + d n + 1) + zw(c + d n + 1)] + F n 2 [wzx( a + c + d + n 2) +yz(b + c + d + n 2) + xy(a + b + d + n 2) + xzy(abcn 2)] + F n 3 [y2w(x(a n + 3)b + c + d)] This relation may made applicable to gauss function 2 F 1 by observing that F (A:1;1;) (C:0;0;) [ (a):(b);(b );) (c):( );( ); x, y] = A+1 F C ((a), b + b ; (c); x) (2.9) It follows now that replacing b + c by b and z by y, we get By replacement we have F n = x n F n ( n, b, (1 a n); y n! x ) (2.10) (a+n)(a+n 1)(a+n 2)x 3 2F 1 ( n 1, b; a n; y/x) = [(ax+by+dw)+n(x+2y+w)](a+n 1)(a+ n 2)x2 2 F 1 ( n, b; 1 a n; y/x)+[xy(2(a n+1)+b)+y 2 (b+1 n)+xw(a+d+1 n)+yw(2(d+1 n)+ b)](a+n 2)nx 2 F 1 ( n+1, b; 2 a n; y/x)+w[xy(2(d+n 2)+b)+y2(b+d+n 2)+xy 2 (a+b+n 2)]n(n 1) 2 F 1 (2 n, b; 3 a n; y/x)+[y 2 w(ax+b+c+d)+x(n 3)]n(n 1)(n 2) 2 F 1 ( n+3, b; 4 a n; y/x). 3 The bilateral generating function U 1 = (1 x/t) a (1 y/t) b (1 z/t) c (1 wt) d = t n G n (3.1) Generating a polynomial form of appell function F 1 proceeding as above G n = By taking partial derivative w.r. to t,we have (c, n)zn F 1 (c + n, a, b; 1 + n; xz, yz) (3.2) n! [( ax/t 2 )(1 y/t)(1 z/t)(1 wt)+( by/t 2 )(1 x/t)(1 z/t)(1 wt)+( cz/t 2 )(1 x/t)(1 y/t)(1 wt) +dw(1 x/t)(1 y/t)(1 z/t)] t n G n = (1 x/t)(1 y/t)(1 z/t)(1 wt) ntn 1G n (3.4) Collect the power of t and equating coefficient of t n, we get wng n + G n+1 [wx(a d n 1) + y(b d n 1) + z(c d n 1) n 1] G n+2 [x(a n 2) +y(b n 2) + z(c n 2) + xyz(a + b d n 2) + xzw(a + c d n 2) 111

4 A. Kumar Pandey: Five-term recurrence relations for hypergeometric functions of the second order +yzw(b + c d n 2)] + G n+3 [xy(a + b n 3) + yz(b + c n 3) + zx(a + c n 3) +zxyw(a + b + c d n 3)] G n+4 [xyz(a + b + c n 4)] = 0 (3.5) replace a + b by a and y by x when G n reduces to G n = (c, n)zn F 1 (c + n, a; 1 + n; xz). (3.6) n! 2 After some algebra (5) becomes 5-term recurrence relation for gauss function 2 F 1 which is c(n + 1)(n + 2)(n + 3) 2 F 1 (c + n, a; 1 + n; xz) + w[x(a 2(n + d + 1) + z(c (n d) n 1)] (c+n)(n+2)(n+3) 2 F 1 (c+n+1, a; 2+n; xz)+[x(a 2(n+2))+z(c (n+2))+x2w(a d n 2) +xzw(a + 2c 2(n d))](c + n)(c + n + 1)(n + 3)z 2 F 1 (c + n + 2, a; 3 + n; xz)[x2(a n 3)+ +xz(a+2c 2(n+3))+x2zw(a (d+n+3))](c+n)(c+n+1)(c+n+2)z2 2 F 1 (c+n+3, a; 4+n; xz) +[x2z(a + c n 4)](c + n)(c + n + 1)(c + n + 2)(c + n + 3)z3 2 F 1 (c + n + 4, a; 5 + n; xz) = 0. 4 Generalizations and conclusion The expressions deduced in the previous sections may readily be generalized to give recurrence relations of any number of terms greater than two by employing generating functions with any given number of binomial factors. It must be pointed out, however, that the complexity of the algebra involved increases rapidly as the number of terms rises. U = (1 X 1 t) a1 (1 X 2 t) a2 (1 X m t) am = t n M n (4.1) (a1, p 1 )(a 2, p 2 ),, (a m, p m ) x p1 1 p 1!p 2!p m!r! xp2 2 xpm m zrt r p1 pm (4.2) Put r = n + p 1 + p p m then the series become (a1, p 1 )(a 2, p 2 ),, (a m, p m ) p 1!p 2! p m!(n + p p m ) xp1 1 xp2 2 xpm m z n+p1+ +pm (4.3) It is clear that M n reduces according to desire replacement of parameters. Proceeding as in previous section many recurrence relation can be obtained after great deal of algebra general hyper geometric function p F q. It is noted that other recurrences for hyper geometric functions exist. Acknowledgements Corresponding Author (A.P.) is thankful to all reviewers and Authors whose references are taken to prepare this article. References 1 Erdelyi A., Higher transcendental functions, McGraw Hill, New York, vol-1(1953). 2 Exton Harold, Four-term recurrence relations for hypergeometric functions of the second order, Collect. Math. 47, 1 (1996), pp Exton H., Multiple hypergeometric functions, Ellis Horwood, Chichester, U.K., (1976). 4 P. Appell et J. Kamp?e de F?eriet, Fonctions hyp?erg?eom?etriques et hyp?ersph?eriques, Gauthier Villars, Paris,

5 South Asian J. Math. Vol. 3 No. 2 5 Srivastava H. M. and Karlsson P. W., Multiple Gaussian hypergeometric series, Ellis Horwood, Chichester, U.K., ( ). 6 Slater L. J., Generalised hypergeometric functions, Cambridge University Press, (1966). 7 Watson G. N., Bessel functions, Cambridge University Press, (1944). 8 Yanez R. J., Dehesa J. S. and Zarzo A., Four-term recurrence relations of hypergeometric type Polynomials, Il nouvo Cimento 109B (1994), pp Rajesh Shrivastava, M. Vijaya Kumar and Mannavi Kohli, Fixed Point Theorem for 2-pairs of Mappings on Menger Space, South Asian J Math, 2012, 2(1), R. Srivastava, S. Agrawal, R. Bhardwaj and R. N. Yadava, Fixed Point Theorems In Complete G-Metric Spaces, South Asian J Math, 2012, 2(2),