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{\bf A COMBINATORIAL PROBLEM THAT AROSE IN BIOPHYSICS}

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{\it By Doron Zeilberger}

Department of Mathematics, Drexel University,
Philadelphia, PA19104.

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

{\bf 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 {\it integers},
$( m_1 , ... . m_s )$, of the system

$$
 m_1 + ... + m_s = 0 ,
\eqno(1a)
$$
$$
 | m_1 | + ... + | m_s | =2m. 
\eqno(1b)
$$          

Then,

$$
  Q( \rho ) := \sum_{m=0}^{\infty} R(m) \rho^{m} = {(1- \rho )}^{-(s-1)}  \sum_{k=0}^{s-1} { (  { s-1 \atop k }  )}^2 \rho^{k}. 
$$

PROOF: It is readily seen that R(m) is the coefficient of 
$\rho^m t^0 $ in

$$
\left ( \sum_{k= - \infty}^{\infty} { t^k  \rho^{|k|/2}} \right )^s = 
{[ \rho^{1/2} t^{-1} / (1- \rho^{1/2} t^{-1} ) + 1 + ( \rho^{1/2} t / (1- \rho^{1/2} t  )] }^s = 
$$
$$
  { ( 1- \rho )}^s {(1- \rho^{1/2} t)}^{-s} {(1- \rho^{1/2} t^{-1} )}^{-s} 
\eqno(2)
$$         

Thus, 
$Q( \rho )$ 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

$$
 Q( \rho ) = {(1- \rho )}^s \sum_{k=0}^{\infty}  
{ (  {s+k-1 \atop s-1}  )}^2 \rho^k 
\eqno(3)
$$       

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

$$
 F(a,b;c;z)= \sum_{n=0}^{\infty} {{ (a)_n (b)_n }\over{ (c)_n n! }} z^{n,} 
\eqno(4)
$$      

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):

$$
 F(a,b;c;z)= (1-z)^{c-a-b} F(c-a,c-b;c;z). 
\eqno(5)
$$          

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

$$
  Q( \rho ) = {( 1 - \rho )}^s F( s,s;1; \rho )= {( 1 - \rho )}^s {( 1- \rho )}^{1-2s} F(1-s,1-s;1; \rho ) = {(1- \rho )}^{-(s-1)} \sum_{k=0}^{s-1} { (  { s-1 \atop k }  )}^2 \rho^k 
$$

                                                            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.,\it 
Effect of fluctuating surface structure and free
energy on the growth of linear tubular aggregates\rm 
{\bf 49}, 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). 
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