4.The ( p,q ) hypergeometric function is defined as where a j R, b l R and for a

Question

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

where aj R, bl 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) = ex,

and if we have the Bessel function Jn(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, ··· , ap)

b =(b1, b2, ··· , bq)

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.

Explanation / Answer

Hypergeometric Function for Numeric and Symbolic Arguments

Depending on its arguments, hypergeom can return floating-point or exact symbolic results.

Compute the hypergeometric function for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

Compute the hypergeometric function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, hypergeom returns unresolved symbolic calls.

Use vpa to approximate symbolic results with the required number of digits:

Special Values

The hypergeometric function has special values for some parameters:

Any hypergeometric function, evaluated at 0, has the value 1:

If, after canceling identical parameters, the list of upper parameters contains 0, the resulting hypergeometric function is constant with the value 1:

If, after canceling identical parameters, the upper parameters contain a negative integer larger than the largest negative integer in the lower parameters, the hypergeometric function is a polynomial. If all parameters as well as the argument x are numeric, a corresponding explicit value is returned:

Hypergeometric functions also reduce to other special functions for some parameters:

Handling Expressions That Contain Hypergeometric Functions

Many functions, such as diff and taylor, can handle expressions containing hypergeom.

Differentiate this expression containing hypergeometric function:

Compute the Taylor series of this hypergeometric function:

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