hypergeon2 http://www.math.kobe-u.ac.jp/OCD/hypergeon2.tfb 2003-08-07 experimental 2002-08-07, 2003-11-30 1 1 hypergeo1 hypergeon0 arith1 fns1 interval1 linalg1 linalg4 relation1 set1 This CD defines symbols for classical hypergeometric series of several variables, which include Appell functions and Lauricella functions. multi_pochhammer multi_pochhammer is a product of pochhammer symbols. $ (a)_n = \prod_{i=1}^{m} (a_{i})_{n_{i}} $ multi_pochhammer(a,n) ~relation1.eq~ arith1.product( interval1.integer_interval(1,linalg4.columnsize(n)) OMLBIND(OMBVAR(i), hypergeo0.pochhammer(linalg1.vector_selector(i,a), linalg1.vector_selector(i,n)))); appel_F1 Appell's hypergeometric series F_1 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 $ F_1(a,b,b',c;x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m+n} (b)_m (b')_n}{(c)_{m+n} (1)_{m} (1)_{n}}x^{m}y^{n}$ apple_F1(a,b1,b2,c,x,y) ~relation1.eq~ arith1.sum( set1.cartesian_product(setname1.N, setname1.N), OMLBIND(OMBVAR(m), arith1.divide(hypergeo0.pochhammer(a,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(b1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(b2,linalg1.vector_selector(2,m)), hypergeo0.pochhammer(c,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(2,m))) ~arith1.times~ ( arith1.power(x,vector_selector(1,m)) ~arith1.times~ arith1.power(y,vector_selector(2,m)) ))); appel_F2 Appell's hypergeometric series F_2 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 $ F_2(a,b,b',c,c';x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_{m}(b')_{n}} {(c)_{m} (c')_{n} (1)_{m} (1)_{n}}x^{m}y^{n} $ apple_F2(a,b1,b2,c1,c2,x,y) ~relation1.eq~ arith1.sum( set1.cartesian_product(setname1.N, setname1.N), OMLBIND(OMBVAR(m), arith1.divide(hypergeo0.pochhammer(a,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(b1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(b2,linalg1.vector_selector(2,m)), hypergeo0.pochhammer(c1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(c2,linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(2,m))) ~arith1.times~ ( arith1.power(x,vector_selector(1,m)) ~arith1.times~ arith1.power(y,vector_selector(2,m)) ))); appel_F3 Appell's hypergeometric series F_3 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 $ F_3(a,a',b,b',c;x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m}(a')_{n}(b)_{m}(b')_{n}} {(c)_{m+n}(1)_{m}(1)_{n}}x^{m}y^{n}$ apple_F3(a1,a2,b1,b2,c,x,y) ~relation1.eq~ arith1.sum( set1.cartesian_product(setname1.N, setname1.N), OMLBIND(OMBVAR(m), arith1.divide(hypergeo0.pochhammer(a1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(a2,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(b1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(b2,linalg1.vector_selector(2,m)), hypergeo0.pochhammer(c,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(2,m))) ~arith1.times~ ( arith1.power(x,vector_selector(1,m)) ~arith1.times~ arith1.power(y,vector_selector(2,m)) ))); appel_F4 Appell's hypergeometric series F_4 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 $ F_4(a,b,c,c';x,y) = \sum_{m,n=0}^{\infty} \frac{(a)_{m+n} (b)_{m+n}}{(c1)_{m}(c2)_{n}(1)_{m}(1)_{n}}x^{m}y^{n} $ apple_F4(a,b,c1,c2,x,y) ~relation1.eq~ arith1.sum( set1.cartesian_product(setname1.N, setname1.N), OMLBIND(OMBVAR(m), arith1.divide(hypergeo0.pochhammer(a,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(b,linalg1.vector_selector(1,m) ~arith1.plus~ linalg1.vector_selector(2,m)), hypergeo0.pochhammer(c1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(c2,linalg1.vector_selector(2,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(1,m)) ~arith1.times~ hypergeo0.pochhammer(1,linalg1.vector_selector(2,m))) ~arith1.times~ ( arith1.power(x,vector_selector(1,m)) ~arith1.times~ arith1.power(y,vector_selector(2,m)) ))); lauricella_FA Lauricella's hypergeometric series F_A of n variables. In case of one variables, it agrees with the Appel function F_2. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: $ F_A(a,b,c;x) = \sum_{k \in \N^n}^{\infty} \frac{(a)_{\sum k_i} \prod (b_i)_{k_i}} {\prod (c_i)_{k_i} \prod (1)_{k_i}} x^{k} $ OMLBIND( OMBVAR(n), lauricella_FA(a,b,c,x) ~relation1.eq~ arith1.sum( hypergeon0.cartesian_product_n(setname1.N, n), OMLBIND(OMBVAR(k), arith1.divide(hypergeo0.pochhammer(a, arith1.sum(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), vector_selector(i,k)))) ~arith1.times~ arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(vector_selector(i,b), vector_selector(i,k)))), arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(vector_selector(i,c), vector_selector(i,k)))) ~arith1.times~ arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(1, vector_selector(i,k))))) ~arith1.times~ hypergeon0.multi_power(x,k) ))) ~hypergeon0.where~ ( n ~relation1.eq~ linalg4.rowcount(b)); lauricella_FC Lauricella's hypergeometric series F_C of n variables. In case of two variable, it agree with the Appel function F_4. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: $ F_C(a,b,c;x) = \sum_{k \in {\bf N}^n}^{\infty} \frac{(a)_{\sum k_i} (b)_{\sum k_i} } {\prod (c_i)_{k_i} \prod (1)_{k_i}} x^{k} $ OMLBIND( OMBVAR(n), lauricella_FC(a,b,c,x) ~relation1.eq~ arith1.sum( hypergeon0.cartesian_product_n(setname1.N, n), OMLBIND(OMBVAR(k), arith1.divide(hypergeo0.pochhammer(a, arith1.sum(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), vector_selector(i,k)))) ~arith1.times~ hypergeo0.pochhammer(b, arith1.sum(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), vector_selector(i,k)))), arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(vector_selector(i,c), vector_selector(i,k)))) ~arith1.times~ arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(1, vector_selector(i,k))))) ~arith1.times~ hypergeon0.multi_power(x,k) ))) ~hypergeon0.where~ ( n ~relation1.eq~ linalg4.rowcount(b)); lauricella_FD Lauricella's hypergeometric series F_D of n variables. In case of two variables, it agree with the Appell function F_1. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: $ F_D(a,b,c;x) = \sum_{k \in {\bf N}^n}^{\infty} \frac{(a)_{\sum k_i} \prod (b_i)_{k_i}} {(c)_{\sum k_i} \prod (1)_{k_i}} x^{k} $ OMLBIND( OMBVAR(n), lauricella_FD(a,b,c,x) ~relation1.eq~ arith1.sum( hypergeon0.cartesian_product_n(setname1.N, n), OMLBIND(OMBVAR(k), arith1.divide(hypergeo0.pochhammer(a, arith1.sum(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), vector_selector(i,k)))) ~arith1.times~ arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(vector_selector(i,b), vector_selector(i,k)))), hypergeo0.pochhammer(c, arith1.sum(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), vector_selector(i,k)))) ~arith1.times~ arith1.prod(interval1.integer_interval(1,n), OMLBIND(OMBVAR(i), hypergeo0.pochhammer(1, vector_selector(i,k))))) ~arith1.times~ hypergeon0.multi_power(x,k) ))) ~hypergeon0.where~ ( n ~relation1.eq~ linalg4.rowcount(b));