Hypergeometric transformations and identities

• DavidSmith
In summary, the conversation discusses the derivation of hypergeometric identities of the form 2F1(a,b,c,z)=gamma function. It is mentioned that hypergeometric functions with values of z other than 1,-1, and 1/2 can also give gamma functions, but the process of deriving them is still unclear. One person shares their experience of using a kummer quadratic identity and kummer transformation to get a hypergeometric function for z=-1/8. The conversation concludes by noting that there are more possible values of z that can give gamma functions, not just 1,-1, and 1/2.
DavidSmith
How do you derive hypergeometirc identities of the form

2F1(a,b,c,z)= gamma function. What I mean is that the hypergeometric function converges to a set of gamme functions function in terms of (a,b,c)

where z is not 1,-1, or 1/2 ?

The hypergeometric identities in the mathworld summary which give gamma functions only have values of 1,-1, and 1/2 for Z but none others.

I have seen hypergeometric functions where z=1/8,1/3, 2/3 etc that give gamma functions but have no idea how to deive them despite many months of time and research.

After spending another 3 days with the problem I used a kummer quadratic identity combined with annother kummer transformation to get a hypergeometric function for z=-1/8

(8^(-3b+1)/2)*2F1((3b-1)/2,-b/2+1/2;1/2+b;-1/8)=

4*(gammafunction(1/2)*gammafunction(2b))/(b*gammafunction(b/2))^2

there are more possible

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Hypergeometric transformations and identities are important tools in mathematics, particularly in the field of special functions. They involve expressing a hypergeometric function in terms of other functions, such as gamma functions. These transformations and identities can be derived using various techniques, such as the binomial theorem, Euler's integral formula, and the Cauchy integral formula.

To derive hypergeometric identities of the form 2F1(a,b,c,z) = gamma function, we can use the binomial theorem. This theorem states that for any real or complex numbers x and y, and any integer n, (x+y)^n can be expanded as a sum of terms of the form x^iy^(n-i), where i ranges from 0 to n. This expansion can then be used to manipulate the hypergeometric function and express it in terms of gamma functions.

For example, let us consider the hypergeometric function 2F1(a,b,c,z). Using the binomial theorem, we can write:

(1+z)^c = ∑(c k)z^k

where (c k) represents the binomial coefficient c choose k. Multiplying both sides by (1-z)^(-b), we get:

(1-z)^(-b)(1+z)^c = ∑(c k)z^k(1-z)^(-b)

Using the definition of the hypergeometric function, we can rewrite the left side as:

2F1(a,b,c,z) = (1-z)^(-b)(1+z)^c

Substituting this into the previous equation, we get:

2F1(a,b,c,z) = ∑(c k)(-b)(-b-1)...(-b-k+1)z^k

= ∑(c k)(-b)^kz^k

= ∑(c k)(-b)^kz^k(k!/(k!)^2)

= ∑(c k)(-b)^k(a)_k/(b)_kz^k/k!

= ∑(c k)(-b)^k(a)_kz^k/k!/(c)_k(b)_k

= (a)_0z^0/(c)_0(b)_0 + ∑(c k)(a)_kz^k/(c)_k(b)_k

= gamma function + ∑(c k)(a)_kz^k

1. What are hypergeometric transformations and identities?

Hypergeometric transformations and identities are mathematical tools used in the study of hypergeometric functions, which are special functions that arise in many areas of mathematics and physics. These transformations and identities allow for the manipulation and simplification of hypergeometric functions, making them easier to solve and analyze.

2. What is the difference between a hypergeometric transformation and a hypergeometric identity?

A hypergeometric transformation is a change of variables that transforms one hypergeometric function into another, while a hypergeometric identity is an equality between two different hypergeometric functions. Transformations are used to simplify a single function, while identities are used to relate different functions to each other.

3. What are some examples of hypergeometric transformations and identities?

One example of a hypergeometric transformation is the Euler transformation, which simplifies a hypergeometric function by reducing the number of terms in its series representation. An example of a hypergeometric identity is the Gauss summation formula, which relates different hypergeometric functions through a linear combination.

4. How are hypergeometric transformations and identities used in practical applications?

Hypergeometric transformations and identities are used in various fields of mathematics and physics, such as probability theory, statistical mechanics, and differential equations. They are also used in computer science for numerical calculations and in engineering for modeling complex systems.

5. Are there any challenges associated with using hypergeometric transformations and identities?

While hypergeometric transformations and identities can simplify and relate hypergeometric functions, there are certain limitations and challenges associated with their use. Some identities may only hold for specific values of the parameters involved, and certain transformations may not always lead to simpler forms of the function. Additionally, the use of these tools requires a strong understanding of hypergeometric functions and their properties.

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