Solving (x*y)^y == gamma((x+2​)*y)/gamma​((x+1)*y) for integer values of x.

Hey all. I'm trying to find the best way to find solutions to an equation (its for island scaling analysis of our quantum dots):

eqn = (x*y)^y == gamma((x+2)*y)/gamma((x+1)*y)

for positive integer values of x.

I was able plot it, and can see solutions are available, and I know at x=1 that y=0.2715. I've tried playing around with solve, vpasolve, functions, and fzero but keep getting errors or empty sym. I know this should be really simple, could someone point me in the right direction or show me how they would code this?

Thank you.

[Walter Roberson]

My tests suggest that in theory there might exist solutions for every positive integer x, but that the solutions quickly approach 1/3, and that numeric precision problems can crop up easily if you try for numeric solutions. Still, it suggests trying fsolve() with initial guess about 0.3

Answer 1

[David Goodmanson]

Hello Christopher,

N = 100;
soln = zeros(1,N)
for x = 1:N
  fun = @(y) (x*y)^y - gamma((x+2)*y)/gamma((x+1)*y);
  soln(x) = fzero(fun,[.1 1])
end
soln

The solution seems to vary over a remarkably small range.

[Walter Roberson]

My tests show you can use an upper bound of 1/3

[David Goodmanson]

Hi Walter,

As additional evidence for your upper bound, the asymptotic formula for the gamma function (Abramowitz and Stegun 6.1.47) gives for the difference between the left and right sides of the equation

 [ gamma((x+2)*y)/gamma((x+1)*y) - (x*y)^y ] / (x*y)^y 

 = (3*y-1)/(2*x) + ... higher order terms in (1/x)    as x -> oo

So for large values of x, setting the leading difference term to be zero means that y = 1/3.