Computer generation of complicated transformations ... - Science Direct

Computer Physics Communications 126 (2000) 141–148 www.elsevier.nl/locate/cpc

Computer generation of complicated transformations and reduction formulas for multiple hypergeometric series A.W. Niukkanen a,1 , O.S. Paramonova b,1 a Vernadsky Institute of Geochemistry and Analytical Chemistry, RAS, Kossygin st. 19, 117 975 Moscow, Russia b Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

Abstract Universal transformation formulas for multiple hypergeometric series obtained recently with the help of operator factorization method have been used to develop an effective software greatly surpassing in this regard all available computer algebra programs. An application of the transformations to derivation of reduction formulas is given. To this end an algorithmic criterion for finding reducible cases was put forward. The criterion holds that any non-trivial reduction formula for a multiple series is a corollary of a trivial (self-obvious) reduction for an appropriately transformed series. The striking effectiveness of the algorithm is illustrated by impressive examples relating to Gel’fand series in 3 and 4 variables connected with the Grassmanians G2,4 and G3,6 , respectively.  2000 Elsevier Science B.V. All rights reserved. Keywords: Multiple hypergeometric series; Computer-aided transformations

1. Introduction Employing canonical forms of multiple series [1] and using the sets of formulas [1] obtained with the help of factorization method [2] linear and quadratic transformations and analytical continuation formulas have been programmed for the most general hypergeometric series. Computer algebra system Maple V, ver. 4 has been used to develop the software. The programs can readily generate those numerous transformations which were referred to in Ref. [3] as a challenge to the theory. Here we confine ourselves to the cases of linear transformations for the Gauss-type canonical forms (see Section 3). The main goal of the present paper is to present theoretical foundation underlying the computer programs and to give convincing evidence of calculational efficiency of the method. To this end we apply the method to derive new reduction formulas for sufficiently complicated functions, such as Gel’fand series in 3 and 4 variables connected with the Grassmanians G2,4 [4,5] and G3,6 [6]. Computer program was being used to generate repeated linear transformations of a given series with respect to different variables and parameters until there appears a series which can be trivially reduced to a simple form by imposing additional conditions on parameters. 1 E-mail: [email protected]. Partially supported by RFBR, Russia.

0010-4655/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 5 1 4 - 7

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A.W. Niukkanen, O.S. Paramonova / Computer Physics Communications 126 (2000) 141–148

2. Notation and definitions Coefficients of a hypergeometric series N F (x1 , . . . , xN ) are fractions with the numerator and denominator formed by products of several Pochhammer symbols of the form (a, I ) = a(a + 1) . . .(a + I − 1) ≡ Γ (a + I )/Γ (a) where I = m · i ≡ m1 i1 + · · · + mN iN . The arbitrary integers [m1 , . . . , mN ] ≡ m will be referred to as spectral components of the parameter a. The whole set of parameters occurring in the symbol (a, I ) will be combined in an elementary list of parameters ha|m1 , . . . , mN i ≡ ha|mi. For brevity we put m ≡ −mn . The empty sets of parameters will be indicated by the symbol ∗. It is implied that (∗, i) ≡ 1. The compound parameter ha|mi where one and only one spectral component is non-zero, say, ha|0, . . . , 0, mn , 0, . . . , 0i, will be called individual parameter. Otherwise it will be called a glueing or “gluon” parameter. The glueing parameters will be placed to the left of the colon (:). The individual parameters will be put to the right of the colon sequentially, according to the positions of their nonzero components. There are N individual positions altogether and contracted symbol ha|mn i with omitted zeroes will be put in nth position instead of the complete symbol. Instead of the symbols hα|1, 1, . . . , 1i and ha|1i with unit spectral components the shorter symbols α and a will be used in the gluon and individual lists, respectively. The symbol d = [a//b] ≡ [a 1, . . . , a A ]//[b1, . . . , bB ] denotes the double set of numerator and denominator parameters specified by identical spectrums (see Eq. (11)). On analogy with m all N component sets will be denoted by vector symbols x ≡ [x1 , . . . , xN ], l ≡ [l1 , . . . , lN ], etc. More complicated sets are denoted as xs ≡ [x1 s1 , . . . , xN sN ], xs m ≡ [x1 s m1 , . . . , xN s mN ], etc. We also use some numbers connected with the iN , i! = i1 ! . . . iN !. The total list of parameters of a hypergeometric sets, i.e. m · i = m1 i1 + · · · + mN iN , xi = x1i1 . . . xN N series F in N variable will be designated as L. In the cases where some variable plays particular rôle we shall use, instead of N F [L; x], differentiated notation N+1 F [L0 , L∗ ; x0 , x] where x0 is the specific variable which is set off from the set of other variables x ≡ [x1 , . . . , xN ]. The list L0 includes compound parameters hα|m0 , mi with m0 6= 0. The parameters of the form hα|0, mi are incorporated into the list L∗ . Occasionally we use one-line notation of the list L. To separate numerator and denominator parameters we use double-slash symbol // in this case. General hypergeometric series in N variables specified by the list of parameters L is defined as a power series of the form   ∞ 1 1 A A X |m i, . . . , hα |m i; x hα N = F [L; x] ≡ N F  L(i)xi /i!, (1) 1 1 B B hβ |l i, . . . , hβ |l i i=0 1 1 1 1 A A (2) L(i) = (α , m · i) . . . (α , m · i) / (β , l · i) . . . (β B , lB · i . Definition 1. Canonical form of the series N+1 F [L0 , L∗ ; x0 , x] with respect to x0 is a form where all negative spectral components connected with x0 are converted into positive form with the help of inversion formula (α, i) = (−1)i (1 − α, −i)−1 where i = m0 i0 + m · i, m0 6 0. Sometimes further reduction to unit spectral components with the help of Gauss–Legendre formula is implied by the definition. Individual parameters occurring in the list L0 of the canonical form should be written as formal glueing parameters (see the last terms in Eqs. (3), (4)). Definition 2. Let p0 be the sum of all numerator spectral components relating to x0 in the corresponding canonical form of the N+1 F and let q0 be the sum of all denominator spectral components. The pair of numbers (p0 //q0 ) will be referred to as a partial type of the N+1 F with respect to x0 . The set [p0 //q0 , p1 //q1 , . . . , pN //qN ] will be called a complete type of the series. To give idea of how to use the above notation and definitions in simple cases we give examples of Appell’s series F1 and Horn series H2 presenting sequentially their traditional designation, explicit expression, our instructive notation and canonical form (see Definition 1) with respect to x1 :


F1 [a, a1, a2 , b; x1, x2 ] =  = F 2

X (a, i1 + i2 ) i1 ,i2

a : a1 ; a2 ; x1 , x2 b : ∗; ∗



(b, i1 + i2 )

(a1 , i1 )(a2 , i2 )

x1i1 x2i2 i1 !i2 !

 = 2 F ha|1, 1i, ha1 |1, 0i : ∗; a2; x1 , x2 , hb|1, 1i : ∗ ; ∗ X

x i1 x i2 (a1 , i1 ) (a2 , i2 )(a20 , i2 ) 1 2 (b1 , i1 ) i1 !i2 ! i1 ,i2     ¯ : a1 ; a2 , a 0 ; x1 , x2 ¯ ha1 |1, 0i : ∗; a2 , a 0 ; x1, x2 hα|1, 1i hα|1, 1i, 2 2  = 2F  . = 2F  ∗ : b1 ; ∗ hb1 |1, 0i : ∗; ∗

H2 [α, a1 , a2 , a20 , b; x1, x2 ] =

143

(3)

(α, i1 − i2 )

(4)

Complete types of all these series are given by the set of numbers [2//1, 2//1].

3. Linear transformations of the Gauss canonical form In the following canonical form of Gauss type (2//1) will be widely used. The form is defined by G=

F hν1 |1, m1 i, hν2 |1, m2 i//hν0 |1, m0 i, L∗ ; x0, x .

N+1

The most important statement concerning the canonical form G is that it satisfies 3 general linear transformations [1] analogous to the Euler transformations of the Gauss F12 series [3]:  F

hν1 |1, m1 i, hν2 |1, m2 i, L∗ ; x0 , x hν0 |1, m0 i

 

= L01 F ≡ (1 − x0 )−ν2   x x0 hν01¯ |1, m01¯ i, hν2 |1, m2 i, hν1 |0, m1 i, L∗ ; , m x0 − 1 (1 − x0 ) 2  ×F  hν0 |1, m0 i, hν01¯ |0, m01¯ i = L02 F ≡ (1 − x0 )−ν1   x x0 , hν1 |1, m1 i, hν02¯ |1, m02¯ i, hν2 |0, m2 i, L∗ ; m 1 x0 − 1 (1 − x0 )  ×F  hν0 |1, m0 i, hν02¯ |0, m02¯ i = L00 F ≡ (1 − x0 )ν012   x hν01¯ |1, m01¯ i, hν02¯ |1, m02¯ i, hν1 |0, m1 i, hν2 |0, m2 i, L∗ ; x0 , ¯ (1 − x0 )m012 . ×F  hν0 |1, m0 i, hν01¯ |0, m01¯ i, hν02¯ |0, m02¯ i

(5)

(6)

(7)

Composition of these transformations provided by our computer program is the most important step toward automatic processing of reduction formulas. The symbols Ln1 , Ln2 and Ln0 will be used to denote transformations with respect to the variable xn and parameters ν1 , ν2 and both of them, respectively.

144


4. Elementary reduction rules and auxiliary nonlinear transformation Analyzing the cases where multiple hypergeometric series assumes the form of a simpler series we first look at those obvious cases where the simplification is due to trivial reasons. In spite of their triviality or rather due to it these cases play, in contrast to traditional methods, very important rôle within our approach. We list these simple cases without proofs accompanying them, if necessary, by appropriate examples. Rule 1 (cancellation of parameters). Equal parameters can be cancelled out from the numerator and denominator list of parameters of a series N F , that is [hα|mi//hα|mi] = [∗//∗]. Rule 2 (contraction of series). If there exists a set of summation indices such that for each compound parameter hα|mi all spectral components corresponding to the set take equal values then contraction of the series takes place on analogy with the example (x234 ≡ x2 + x3 + x4 ): 4 F hd1 |1333i, hd2 |2111i, . . .; x1, x2 , x3 , x4 = 2 F hd1 |13i, hd2 |21i; x1, x234 . (8) Rule 3 (factorization of series). The set of indices numerating non-zero spectral components mn will be called support of the spectrum m = [m1 , . . . , mN ]. The union of supports of all spectra entering a group of spectra m, l, q, etc. will be called support of the group. If the parameters occuring in the series N F can be broken down into groups with non-intersecting supports then the series is broken down into a product of the series corresponding to the groups. The representative example is 3 F hd1 |m1 m2 0i, hd2 |00li; x1, x2 , x3 = 2 F hd1 |m1 ; m2i, x1 , x2 1F hd2 |li; x3 . (9) Rule 4 (suppression of variables). If a series N F contains the parameter h0|mi where mn > 0, n = 1, . . . , N then the summation indices of the N F corresponding to positive mn ’s vanish. The corresponding variables and spectral components should be cancelled out. Only those quantities connected with zero mn ’s survive in the N F . Rule 5 (binomial reduction). The series b ≡ N+1 F (x0 , x) having binomial type (1//0) with respect to x0 is expressed through the series in N variables. An explicit formulation is (10) b ≡ N+1 F hν|1, mi, L∗ ; x0, x = (1 − x0 )−ν NF hν|1, mi, L∗ ; (1 − x0 )−m x . ¯ pi with Rule 6 (auxiliary transformation). Consider the series N+1 F containing an “indefinite” parameter h0|1, ¯ pi where 1¯ ≡ −1, p1 > 0, . . . , pN > 0. To reduce the series to a simpler zero argument and alternating spectrum |1, form containing but “definite” parameters introduce new variable j0 = i0 − p1 i1 − · · · − pN iN instead of i0 . ¯ pi is (0, −j0 ) from which we have j0 > 0. We also note that The Pochhammer symbol corresponding to h0|1, (0, −j0 ) = (−1)j0 /j0 !. With all these remarks in mind we readily conclude that ¯ pi, hd0 |m0 , mi, hd0 |l0 , li, . . . : d0 ; d1 ; . . . ; dN ; x0, x F h0|1, 1 2 = N+1F hd0 //1|1, pi, hd01 |m0 , m + m0 pi, hd02 |l0 , l + l0 pi, . . . : p p ∗; d1 ; d2 ; . . . ; dN ; −x0, x1 x0 1 , . . . , xN x0 N .

N+1

(11)

The trivial reduction rules given above present the second important step of our algorithm aimed at finding of reducible cases of a given hypergeometric series.


145

5. Reduction formulas for Gel’fand functions An interesting class of hypergeometric functions connected with the Grassman manifold Gk,n whose elements are k-dimensional subspaces of an n-dimensional linear space C n was introduced in [4–6]. The cases of G2,4 [5] and G3,6 [6] recieved the most study. The interesting result concerning reduction formulas for the Gel’fand functions connected with Gk,n is that they allow, under certain conditions, different realizations in the form of two series of multiplicities (k − 1)(n − 1) and (k − 1)(n − k − 1). The only Gel’fand functions connected with G2,4 is [5] ¯ ¯ : a1 ; a2 ; a3 ; x1 , x2 , x3 . 1, 1i (12) p(a, b; x) ≡ 3 F ha|1, 1, 1i//hb|1, Three Gel’fand functions connected with G3,6 are [6] Φ1 (x) = 4 F ha1 |1100i, ha2|0011i, ha3|1010i, ha4|0101i//hb|1111i; x , ¯ ha5 |1¯ 101i; ¯ x, Φ2 (x) = 4 F ha1 |1100i, ha2|0011i, ha3|0110i, ha4|101¯ 1i, ¯ ¯ 11i, ¯ ha5 |1111i; ¯ x, ha4 |10 Φ4 (x) = 4 F ha1 |1100i, ha2|1010i, ha3|1¯ 101i,

(13) (14) (15)

where x = [x1 , x2 , x3 , x4 ], and three others insignificantly differ from those given above. The only reduction formula obtained in [5] for p(x) connects p(1 − a3 , a1 + a2 ; x) with the Gauss series F12 (z) where z is rational function of x1 , x2 , x3 . Less interesting formulas connecting Φ1 , Φ2 , Φ4 with the series in 10 (!) variables can be obtained from a general formula [4]. To compare two approaches to the theory of hypergeometric functions we briefly run through reducibility properties of the functions p, Φ1 , Φ2 and Φ4 . Theorem 1. If b = a12 ≡ a1 + a2 then p(x) reduces to the Horn functions H2 , G2 and to the Appel function F2 . Proof. Transforming p(x) to pa (x) = L12 L31 p(x) we put b = a12 and then apply Rules 1 and 2 to the triple series pa (x). The result is p(a, a12 ; x) = (1 − x1 )−a (1 − x3 )−a3 × H2 a, a2, a3 , a12 − a, a12; (x2 − x1 )/(1 − x1 ), x3 (1 − x1 )/(1 − x3 ) .

(16)

Using appropriate canonical forms of H2 (see Eq. (4)) we first apply L11 to H2 and then use the Rule 1 which leads to F2 series: p(a, a12 ; x) = (1 − x1 )a2 −a (1 − x2 )−a2 (1 − x3 )−a3 × F2 a12 − a, a2, a3 , a12 , 1 − a; (x1 − x2 )/(1 − x2 ), x3 (x1 − 1)/(1 − x3 ) . Applying L22 to the H2 and employing the same Rule 1 we obtain the case of G2 function [3].

(17) 2

Theorem 2. If a = 1 − a3 then p(x) reduces to the Appell’s series F1 and F3 . Proof. Transforming p(x) to pb (x) = L31 L21 L12 p(x), putting a = 1 − a3 we then apply “suppression” Rule 4 to obtain reduction of p(1 − a3 , b; x) to the Appell’s F1 . Using canonical form of the F1 (see Eq. (3)) and applying the transformation L11 we obtain reduction to F3 . 2 Theorem 3. p(1 − a3 , a12 ; x) reduces to F12 . Proof. Let a = 1 − a3 in Eq. (17). Due to the Rule 1 binomial Rule 5 is applicable to the new series. The final result

146


p(1 − a3 , a12; x) = (1 − x1 )a3 −1 (1 − x3 )a12 −1 (1 − x1 x3 )1−a13 (1 − x2 x3 )−a2 × F12 1 − a3 , a2 //a12 ; (x2 − x1 )(1 − x3 )/(1 − x1 )(1 − x2 x3 )

(18)

is equivalent to Gel’fand, Graev and Retakh reduction formula [4,5]. 2 Theorem 4. If b = a1 + a2 ≡ a12 then Φ1 reduces to F1 and F3 . If b = a12 = a34 then Φ1 reduces to F12 . Proof. Introducing Φ1a = L21 L11 Φ1 and using Rule 1 we can apply Rule 4 twice. This gives us reduction of Φ1 to F1 . In turn L11 F1 takes the form of F3 . If a12 = a34 then, after transformation L22 of the F1 , Rules 1 and 2 can be applied, consecutively, to give the desired reduction of the Φ1 to F12 . 2 Theorem 5. If a345 = 1 then Φ2 reduces to the Appell F2 . If a24 = 1 the Φ2 reduces to F1 . Proof. Introducing the condition a3 + a4 + a5 ≡ a345 = 1 into Φ2a = L42 L12 Φ2 and using Rule 1 we can apply twice the Rule 2 to the transformed series to obtain the F2 reduction. Introducing a24 = 1 in Φ2a and using Rule 1 we obtain the series Φ2b having binomial type (1//0) with respect to the fourth argument. Employing Rule 5 and ¯ Using Rule 6 we can applying L31 to the resultant 3 F series we get a new series 3 F with the parameter h0|101i. 2 1 apply then Rule 2 to obtain a non-Hornian F series. Applying L1 to this series we finally get the desired reduction to F1 . 2 We note without proof that under conditions a35 = 1, a24 = 1 and a35 = a31 = 1 the function Φ4 reduces to Appell’s F2 , Horn’s G2 and Gauss’ F12 , respectively. Out of 13 reduction formulas given above only one (see Eq. (18)) was known earlier. We conclude by important remark that we give not so much the formulas as an effective algorithm which can be readily applied to any series satisfying necessary conditions.

6. Exposition of Maple input and outputs An automated version of our program is still being developed now. At present we are content with an interactive version of the program. As an example of Maple input and outputs typical for the latter program we give here the computer proof of our Theorem 2. The notation used in the session given below is as follows: nx is the number of variables, ni is the set of individual numerator parameters (initially, all individual denominator sets are empty), ng is the list of glueing numerator parameters, dg is the list of glueing denominator parameters, and lx is the set of variables. All other symbols are almost self-explanatory with the possible exception of the symbol C which relates to analytical continuation formulas. Each cycle of the procedure is finished with explicit expression of a currently transformed series and the chain of the transformations leading to the series (the two-line symbols containing only numbers are complete types of the series). The two series arisen in the fourth cycle are just those Appell’s F1 and F3 which occur in the formulation of the Theorem 2. The proof of the Theorem 2 for the series p(x) runs as follows. > restart; > read`c : \\ mywork\\ main.txt`; > main(): > nx : 3 > ni : a1; a2; a3; > ng : ha|1, 1, −1i > dg : hb|1, 1, −1i


> lx : x1, x2, x3 >] [2 2 [1 1

2] 1]

Would you like to continue (y–yes, n–no)? >y 1 : L1, L2, L0, C; 2 : L1, L2, L0, C; 3 : L1, L2, L0, C Indicate transformation number (for example – 1L0, where 1 is variable number): > 1L2 You can do some assignment. For example: > a := s − 1; b : w − 17; > F [ha|1, 1, −1i : a1; a2; a3; x1, x2, x3] [hb|1, 1, −1i : ∗ ; ∗ ; ∗ ] (1 − x1)ˆ(−a1)∗ ∗F [ha|0, 1, −1i : a1, b − a; a2; a3; x1/(x1 − 1), x2, x3] [hb|1, 1, −1i : ∗ ;∗ ; ∗ ] [2 2 [1 1

2] [L2] [2 2 2] 1] [ 1 ] [1 1 1]

Would you like to continue (y–yes, n–no)? >y 1 : L1, L2, L0, C; 2 : L1, L2, L0, C; 3 : L1, L2, L0, C Indicate transformation number: > 2L1 You can do some assignement: > (1 − x1)ˆ(−a1)∗(1 − x2)ˆ(−a2)∗ ∗F [hb − a|1, 1, 0i : a1; a2; a3 ; x1/(x1 − 1), x2/(x2 − 1), x3] [ hb|1, 1, −1i : ∗ ; ∗ ; 1 − a ] [2 2 [1 1

2] [L2] [2 2 2] [L1] [2 2 2] 1] [ 1 ] [1 1 1] [ 2 ] [1 1 1]

Would you like to continue (y–yes, n–no)? >y 1 : L1, L2, L0, C; 2 : L1, L2, L0, C; 3 : L1, L2, L0, C Indicate transformation number: > 3L1 You can do some assignement: > −(1 − x1)ˆ(−a1)∗(1 − x2)ˆ (−a2)/(−1 + x3)∗ (1 − x3)ˆb∗ ∗F [h1 − b| − 1, −1, 1i, hb − a|1, 1, 0i : a1; a2; 1 − a − a3; [ ∗ : ∗; ∗ ; 1 − a x1 ∗ (−1 + x3)/(x1 − 1), x2 ∗ (−1 + x3)/(x2 − 1), x3/(−1 + x3)] ]

147

148

[2 2 [1 1


2] [L2] [2 2 2] [L1] [2 2 2] [L1] [2 2 2] 1] [ 1 ] [1 1 1] [ 2 ] [1 1 1] [ 3 ] [1 1 1]

Would you like to continue (y–yes, n–no)? >y 1 : L1, L2, L0, C; 2 : L1, L2, L0, C; 3 : L1, L2, L0, C Indicate transformation number: > 1L1 You can do some assignement: > a := 1 − a3; −(1 − x1)ˆ(−a1)∗(1 − x2)ˆ (−a2)/(−1 + x3)∗ (1 − x3)ˆb∗ ∗F [b − 1 + a3 : a1; a2; −x1 ∗ (−1 + x3)/(x1 − 1), −x2 ∗ (−1 + x3)/(x2 − 1)] [ b : ∗; ∗ ] It is Gauss-type [2 2] [1 1] −(1 − x2)ˆ(−a2)/(−1 + x3)∗ (1 − x3)ˆ b∗ (1 − x1∗ x3)ˆ(−a1)∗ ∗F [∗ : 1 − a3, a1; b − 1 + a3; a2; x1 ∗ (−1 + x3)/(−1 + x1 ∗ x3), −x2 ∗ (−1 + x3)/(x2 − 1)] [b : ∗ ; ∗ ] [2 2 [1 1

2] [L2] [2 2 2] [L1] [2 2 2] [L1] [2 2 2] 1] [ 1 ] [1 1 1] [ 2 ] [1 1 1] [ 3 ] [1 1 1]

[ some ] [2 2] [L1] [2 2] [ assign ] [1 1] [ 1 ] [1 1] Would you like to continue (y–yes, n–no)? >n Bye!!! > References [1] A.W. Niukkanen, Transformation theory of multiple hypergeometric series and computer aided symbolic calculations, in: Proc. 9th Conference on Computational Modelling and Computing in Physics, Dubna, 1997, pp. 219–223. [2] A.W. Niukkanen, A new method in the theory of the hypergeometric series, Uspekhi Mat. Nauk 43 (1988) 191–193. [3] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions 1 (McGraw-Hill, New York, 1953). [4] I.M. Gel’fand, M.I. Graev, V.S. Retakh, General hypergeometric equations systems and hypergeometric type series, Uspekhi Mat. Nauk 47 (1992) 3–82 (in Russian). [5] I.M. Gel’fand, M.I. Graev, V.S. Retakh, Reduction formulas for hypergeometric functions on the Grassmanian Gk,n , Dokl. Acad. Sci. USSR 318 (1991) 793–797. [6] I.M. Gel’fand, M.I. Graev, V.S. Retakh, Hypergeometric functions connected with the Grassmanian G3,6 , Mat. Sb. 180 (1989) 3–38.