Question

4.The (p,q) hypergeometric function is defined as

where a_j ∈ R, b_l ∈ R and for any real value a we have that

(a)₀ =1

(a)_n =a(a + 1)(a + 2)···(a + n − 1), n ≥ 1.

So for example, we have that

0F₀(;;x) = e^x,

and if we have the Bessel function J_n(x) where

,

then we have

.

Write a program which computes the (p,q) hypergeometric function.

It should take as input vectors a and b where

a =(a₁, a₂, ··· , a_p)

b =(b₁, b₂, ··· , b_q)

and an evaluation point x. Note, your program should determine p and q using the length of a and b respectively. Make sure you vectorize your function. Make sure you make efficient use of recursion. Clearly explain the stopping criteria you choose and why you choose it. Test your code by comparing your results to those you would get using the “bessel plotter” code. Provide plots to explain your validation.

Then, using Matlab’s version of the functions as the “true” values, numerically prove the identities

Generate semi-log plots of the difference of the two functions over meshes on [−.5,.5] and argue why this shows the identities are true. Note, you may need to adapt your stopping condition to adequately answer this problem. Why does the validity of the series break down as x gets close to ±1? Again, provide plots to justify your answer and make sure to say something about the domain of definition of the functions you are trying to approximate.