GAUSS’S _{2}F_{1}(1) CANNOT BE GENERALIZED TO _{2}F_{1}(x)

Doron Zeilberger *

Appeared in: J. Comp. Appl. Math. 39 (1992), 379-382.

Abstract: Using ideas of Jet Wimp and Richard McIntosh, it is proved that Gauss’s explicit evaluation of _{2}F_{1}(a, b; c; 1) cannot be generalized to _{2}F_{1}(a, b; c; x), for arbitrary a, b, c, and x. A short proof of Wimp’s theorem that asserts that _{3}F_{2}(a, b, c; d, e; 1) cannot be expressed in closed form is also given.

0 Introduction

As we all know, a geometric series is a series

∑_{k=0}^{∞} A_{k}  (1)

such that the ratio A_{k+1}/A_{k} of consecutive terms is identically equal to a constant, say r, and if r < 1, such a series is explicitly summable, the sum being A_{0}/(1 − r).

A hypergeometric series (e.g. [R], [B]) is a series (1) where A_{k+1}/A_{k} is a rational function of k. Writing this rational function in factored form

\frac{x(k + a_{1})(k + a_{2})...(k + a_{p})}{(k + 1)(k + b_{1})(k + b_{2})...(k + b_{q})} ,  (2)

the resulting series (1) (with A_{0} = 1) is denoted by

pF_{q}({a_{1} , ... , a_{p} \atop b_{1} , ... , b_{q}} ; x) .

(The extra factor (k + 1) in the denominator of (2) is there for historical reasons, of course it is possible to get rid of it by making one of the numerator parameters a_{i} equal to 1.) In terms of the raising factorial (a)_{k} := a(a + 1)...(a + k − 1), the general hypergeometric series can be written, and is usually defined by

pF_{q}({a_{1} , ... , a_{p} \atop b_{1} , ... , b_{q}}; x) = ∑_{k=0}^{∞} \frac{(a_{1})_{k}...(a_{p})_{k}}{k!(b_{1})_{k}...(b_{q})_{k}} x^{k} .

There are many cases in which a hypergeometric series can be summed ”explicitly”, i.e. in terms of powers and products of Gamma functions. The simplest case is the binomial theorem:

1F_{0}({a \atop −} ; x) = (1 − x)^{−a} .  (3)

* Department of Mathematics, Temple University, Philadelphia, PA19104. Supported in part by NSF grant DMS8800663.

Moving up to _{2}F_{1}, we have Gauss’s celebrated formula(e.g. [R], theorem 18, p.49):

2F_{1}({a, b \atop c}; 1) = \frac{Γ(c)Γ(c − a − b)}{Γ(c − a)Γ(c − b)} .  (4)

Despite many attempts, no such formula for F (a, b; c; x), with general x and general a, b, c was ever found, and the first purpose of the present paper is to prove that such a formula is impossible. Of course, for various specializations of x and a, b, c there do exist closed form formulas([B],[G-S]), the most well known being that of Kummer, in which x = −1 and c = a − b + 1([R], Theorem 26, p. 68):

2F_{1}({a, b \atop 1 + a − b} ; −1) = \frac{Γ(1 + a − b)Γ(1 + a/2)}{Γ(1 + a/2 − b)Γ(1 + a)} .  (5)

Another natural generalization of Gauss’s formula would be an explicit expression for the general

3F_{2}({a_{1}, a_{2}, a_{3} \atop b_{1}, b_{2}} ; 1) .  (6)

The celebrated formula of Pfaff-Saalschutz sums (6) ([R], p. 87) when one of the numerator parameters is a negative integer (so the series is terminating) and in addition it is balanced: a_{1} + a_{2} + a_{3} + 1 = b_{1} + b_{2}. This gives a four-parameter formula. The formula of Dixon([R], p. 92) sums (6) when it is well-poised: a_{1} + 1 = b_{1} + a_{2} = b_{2} + a_{3}, giving a 3-parameter formula. Other 3-parameter formulas are associated with the names of Watson and Whipple (see [B]). In addition there are many ”strange” 2- and 1-parameter formulas conjectured by Gosper and proved by Gessel and Stanton[G-S]. The impossibility of a closed form evaluation of a general 3F_{2}(a, b, c; d, e; 1) is a remarkable result due to Wimp[W]. Wimp used recurrences and an ingenious asymptotic argument. The second purpose of this paper is to give a short proof of Wimp’s theorem. My proof uses Wimp’s beautiful ideas, but the details are much shorter. I was also very much influenced by McIntosh’s brilliant thesis[M] and his approach to proving minimality of recurrences.

1 _{2}F_{1}(a, b; c; x) Is Not Nice

Theorem: There is no formula of the form

2F_{1}(a, b; c; x) = Kλ^{A_{0}a+B_{0}b+C_{0}c} \frac{∏_{j=1}^{p} Γ(A_{j}a + B_{j}b + C_{j}c + D_{j})}{∏_{j=1}^{q} Γ(A′_{j}a + B′_{j}b + C′_{j}c + D′_{j})} ,  (7)

with A_{0} , A_{1} , ... , A_{p} , B_{0} , B_{1} , ..., B_{p} , C_{0} , C_{1} , ... , C_{p ,} A′_{1} , ... , A′_{q} , B′_{0} , B′_{1} , ... , B′_{q} , C′_{1} , ... , C′_{q} all rational numbers, and K and λ are allowed to depend on x.

Proof: Suppose there is such a formula, then there would be a formula for

a(n) := _{2}F_{1}(−n, n + 1; 1; −1) = ∑_{k=0}^{n}({n + k \atop k})({n \atop k})  (8)

of the form

a(n) = Kλ^{A_{0}n} \frac{∏_{j=1}^{p} Γ(A_{j}n + D_{j})}{∏_{j=1}^{q} Γ(A′_{j}n + D′_{j})} ,  (9)

with A_{0}, A_{j}, and A′_{j} rational numbers. Let K be the least common denominator of A_{0}, A_{j}, A′_{j}.Then obviously a(n + K)/a(n) would be a rational function of n, i.e., there would be polynomials P (n) and Q(n) such that

P (n)a(n + K) − Q(n)a(n) ≡ 0, n = 0, 1, 2... .  (10)

Note that P (n) and Q(n) can be chosen to have integer coefficients, since a(n) are integers, and writing P (n) and Q(n) in generic form, plugging in the first few values of n, and solving the resulting system of linear equations, gives rational, and hence integer, solutions.

A routine application of the method of [K], pp. 66-67, shows that a(n + 1)/a(n) tends to (1 + \sqrt{2})^{2}, as n → ∞ (see also [P], p. 202). This implies that a(n + K)/a(n) tends to (1 + \sqrt{2})^{2K} as n → ∞. On the other hand if (10) were true, it would follow that if a(n + K)/a(n) tends to any non-zero limit, that limit must be a rational number. Now (1 + \sqrt{2})^{2K} always has the form a + b\sqrt{2}, with a and b integers, and b ≠ 0 (use the binomial theorem), and this must be irrational, thanks to Hippasus of Metapontum (see also [W’]).

2 A Short Proof of Wimp’s Theorem

I will now give a short proof of Wimp’s theorem[W] that _{3}F_{2}(a, b, c; d, e; 1) cannot be expressed in closed form, in the form analogous to (7). If it were then

b(n) := _{3}F_{2}(−n , −n , n + 1 ; 1 , 1 ; 1) = ∑_{k=0}^{n}({n + k \atop k})({n \atop k})^{2}  (11)

would have a formula of the form (9), which would entail, for some integer K, and polynomials P (n), Q(n) with integer coefficients that

P (n)b(n + K) − Q(n)b(n) ≡ 0, n = 0, 1, 2... .  (12)

A routine application of the method of [K], pp. 66-67, shows that b(n + 1)/b(n) tends to ((1 + \sqrt{5})/2)^{5}, as n → ∞ (see also [P], p. 200). The rest of the proof goes ditto, with \sqrt{2} replaced by \sqrt{5}, and 2K replaced by 5K.

Epilogue

The referee has found a much quicker proof of the main result of this paper. He pointed out that the specialization

2F_{1}(a, a + 1/2; 1/2; x) = (1/2)(1 + \sqrt{x})^{−2a} + (1/2)(1 + \sqrt{x})^{2a}

also immediately implies that Gauss’s evaluation of _{2}F_{1}(1) cannot be extended to _{2}F_{1}(x).

In spite of this beautiful proof of the referee, the present paper still serves a purpose. I believe that, although the referee calls his or her proof ”trivial”, it is very sleek, and I am glad that I can use the present paper to present it. Moreover, trivial or not, the result stated in the title is interesting, and ought to be pointed out. Finally, the method described in this paper, that was inspired by McIntosh and Wimp, can be used for proving non-closed-formness of many other hypergeometric and combinatorial sums.

REFERENCES

[B] Bailey, W. N. , ”Generalized Hypergeometric Series”, Cambridge Math. Tracts 32, Cambridge University Press, London, 1935. (Reprinted: Hafner, New York, 1964 .)

[G-S] Gessel, I., and Stanton, D., Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), 295-308.

[K] Knuth, D. E., ”Sorting and Searching”, The art of computer programming, vol. 3, Addison-Wesley, Reading, 1973.

[M] McIntosh, R. J., Asymptotic and Arithmetic Properties of Recurrent Sequences, Ph.D. dissertation, UCLA, 1989.

[P] van der Poorten, A., A Proof that Euler missed..., Apéry’s proof of the irrationality ofζ(3), Math. Intel. 1(1979), 195-203.

[Ra] Rainville, E. D., ”Special Functions”, Chelsea, Bronx, 1971. Originally published by Macmillan, 1960.

[W] Wimp, J., Irreducible recurrences and representation theorems for_{3}F_{2}(1), Comp. & Maths. with Appl. 9(1983), 669-678.

[W’] Wimp, J,.The square root of two is irrational, in: ”Against Infinity”, an anthology of contemporary mathematical poetry, (edited by E. Robson and J. Wimp), Primary Press, Parker Ford, PA, p. 79.

May 1990. Revised: Oct, 1991.