A COMBINATORIAL PROBLEM THAT AROSE IN BIOPHYSICS By Doron Zeilberger

Department of Mathematics, Drexel University, Philadelphia, PA19104.

Current address (1990-): Dept. of Mathematics, Temple University, Philadelphia, PA 19122.

Appeard in: Fibonacci Quarterly 27 (1989), 372.

The purpose of this note is to prove the following result that was conjectured by T.L.Hill ([1], [2], p.148) in the course of his investigations of the ”surface” properties of some long multi-stranded polymers.

THEOREM: Let s be a positive integer, and for any non-negative integer m, let R(m) be the number of solutions, in integers, (m_{1}, ....m_{s}), of the system

m_{1} + ... + m_{s} = 0,  (1a)
|m_{1}| + ... + |m_{s}| = 2m.  (1b)

Then,

Q(ρ) := ∑_{m=0}^{∞} R(m)ρ^{m} = (1 − ρ)^{−(s−1)} ∑_{k=0}^{s−1} ({s − 1 \atop k})^{2}ρ^{k}.

PROOF: It is readily seen that R(m) is the coefficient of ρ^{m}t^{0} in

( ∑_{k=−∞}^{∞} t^{k}ρ^{|k|/2})^{s} = [ρ^{1/2}t^{−1}/(1 − ρ^{1/2}t^{−1}) + 1 + (ρ^{1/2}t/(1 − ρ^{1/2}t)]^{s} =

(1 − ρ)^{s}(1 − ρ^{1/2}t)^{−s}(1 − ρ^{1/2}t^{−1})^{−s}  (2)

Thus, Q(ρ) is the coefficient of t^{0} in the right side of (2). Expanding the last two terms in the right side of (2) by Newton’s binomial formula, and collecting the coefficient of t^{0} we get

Q(ρ) = (1 − ρ)^{s} ∑_{k=0}^{∞} ({s + k − 1 \atop s − 1})^{2}ρ^{k}  (3)

Recall that the hypergeometric function F(a,b;c;z) is defined by ( e.g. [3] chapter 4):

F (a, b; c; z) = ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}n!} z^{n,}  (4)

where (a)_{n} := a(a + 1)...(a + n − 1).

One of the most celebrated formulas in the theory of hypergeometric functions is Euler’s transformation ([3], theorem 21, p.60):

F (a, b; c; z) = (1 − z)^{c−a−b}F (c − a, c − b; c; z).  (5)

Converting the right side of (2) into hypergeometric notation, using (5) , and converting back yields:

Q(ρ) = (1 − ρ)^{s}F (s, s; 1; ρ) = (1 − ρ)^{s}(1 − ρ)^{1−2s}F (1 − s, 1 − s; 1; ρ) = (1 − ρ)^{−(s−1)} ∑_{k=0}^{s−1} ({s − 1 \atop k})^{2}ρ^{k}

Q.E.D. The same method of proof can be applied to treat the more general problem where the 0 at the left side of (1a) is replaced by a general integer i. REFERENCES

1. Hill,T.L.,Effect of fluctuating surface structure and free energy on the growth of linear tubular aggregates49, 1017-1031 (1986).

2. Hill,T.L.,”Linear Aggregation Theory in Cell Biology”, Springer-Verlag, New York, (1987).

3. Rainville,Earl D., ”Special Functions”, Chelsea, Bronx, New York, (1971).