Understanding Notation and Connections of Hypergeometric Functions

In summary: There is no easy way to see in which interval a given function or family of functions converge. One must calculate the convergence for each individual function.
  • #1
matematikuvol
192
0
Hypergeometric function is defined by:
[tex]_2F_1(a,b,c,x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n[/tex]
where ##(a)_n=a(a+1)...(a+n-1)##...
I'm confused about this notation in case, for example, ##_2F_1(-n,b,b,1-x)##.
Is that
[tex]_2F_1(-n,b,b,1-x)=\sum^{\infty}_{n=0}\frac{(-n)_n}{n!}(1-x)^n[/tex]
or
[tex]_2F_1(-n,b,b,1-x)=\sum^{\infty}_{k=0}\frac{(-n)_k}{k!}(1-x)^k[/tex]
and how to summate ##_2F_1(-n,b,b,1-x)##?

And one more question. Are the generalised hypergeometric function and confluent hypergeometric function same function?
 
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  • #2
Hi !

One must not use the same symbol for two different parameter, index or variable.

[tex]_2F_1(a,b,c,x)=\sum^{\infty}_{k=0}\frac{(a)_k(b)_k}{k!(c)_k}x^k[/tex]
where ##(a)_k=a(a+1)...(a+k-1)##

[tex]_2F_1(-n,b,b,1-x)=\sum^{\infty}_{k=0}\frac{(-n)_k}{k!}(1-x)^k[/tex]

In case of negative parameter (-n), the Hypergeometric2F1 function reduces to a Jacobi polynomial. Moreover, with the two other equal parameters (b), it reduces to an even simpler function :
[tex]_2F_1(-n,b,b,1-x)=x^n[/tex]

The confluent hypergeometric functions are degenerate hypergeometric functions. Also known as Kummer and Tricomi functions.
See Table 9, page 36 in the paper "Safari in the contry of special functions" :
http://www.scribd.com/JJacquelin/documents
 
  • #3
Thanks a lot for your answer. One more short question.

"The confluent hypergeometric functions are degenerate hypergeometric functions."
Why? I don't see that.
Confluent hypergeometric function is ##F(a;b;x)## and hypergeometric function is ##F(a,b;c,x)##. Why ##F(a;b;x)## is called degenerate hypergeometric function?
 
  • #4
Check out the connection between the two. It involves a limit on one of the parameters.
 
  • #5
matematikuvol said:
Thanks a lot for your answer. One more short question.

"The confluent hypergeometric functions are degenerate hypergeometric functions."
Why? I don't see that.
Confluent hypergeometric function is ##F(a;b;x)## and hypergeometric function is ##F(a,b;c,x)##. Why ##F(a;b;x)## is called degenerate hypergeometric function?

No, F(a;b;x) is not called "degenerate hypergeometric function". That is not what I mean. "degenerate" is not the name of a function. It is a manner to say that a function of higher level tends and reduces to another function of lower level when one or several parameters tend to some particular values or some particular relationship.
By the way, the correct name of F(a,b;c,x) is not "hypergeometric function", but is the "Gauss Hypergeometric function" or the "Hypergeometric2F1 function".
A name for F(a;b;x) is "Hypergeometric1F1" or a "confluent hypergeometric function".
Hypergeometric functions is a more general name for a very large family of functions, including Hypergeometric1F1, Hypergeometric2F1 and many others. This is explained in the paper "Safari in the contry of the special functions", pages 26, 27, 28 with a large number of examples of various hypergeometric functions :
http://www.scribd.com/JJacquelin/documents
 
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  • #6
Tnx a lot for usefull answer. All hypergeometric function converge for ##|x|<1##. Right? So for example
[tex]\ln (1+x)=x_2F_1(1,1;2;-x)[/tex]
This is correct for which ##x##? Only for ##|x|<1##. Right? I don't understand jet why those functions are so important. Ok for example I want to know Legendre polynomials in term of hypergeometric series. I know that
[tex]\frac{1}{\sqrt{1-2xt+t^2}}=\sum^{\infty}_{k=0}P_k(x)t^k[/tex]
What is easiest way to get relationship beetween ##P_n## and some of the family of hypergeometric functions.
 
  • #7
The expression of Legendre Polynomial in term of Hypergeometric function is given p.27 in the paper referenced in my preceeding post.
I don't know if there is an easier way to get relationship between a given function and a particular hypergeometric function than expending the given function into various forms of series and check if the pattern of one of them corresponds to the general pattern of the hypergeometric series. I don't think that a systematic method exists.
 
  • #8
matematikuvol said:
Tnx a lot for usefull answer. All hypergeometric function converge for ##|x|<1##. Right? So for example
[tex]\ln (1+x)=x_2F_1(1,1;2;-x)[/tex]
This is correct for which ##x##? Only for ##|x|<1##. Right? I don't understand jet why those functions are so important. Ok for example I want to know Legendre polynomials in term of hypergeometric series. I know that
[tex]\frac{1}{\sqrt{1-2xt+t^2}}=\sum^{\infty}_{k=0}P_k(x)t^k[/tex]
What is easiest way to get relationship beetween ##P_n## and some of the family of hypergeometric functions.

No. Kummer function converge for all ##x##.
 
  • #9
?

I don't understand why.

J. Jacquelin
I didn't get the answer from reading your text. How you get connection between Legendre polynomial and #_2F_1#?
 
  • #10
#_2F_1(a,b;c;x)# converge for #|x|<1#. Ok so I know that from this hypergeometric function I could define Legendre polynomials because they are defined for #|x|<1#. From Safari file
[tex]P_n(x)=_2F_1(-n,n+1;1;\frac{1-x}{2})[/tex]
From this table I see that
[tex]T_n(x)=_2F_1(-n,n;\frac{1}{2};\frac{1-x}{2})[/tex]
Laquerre polynomials are defined on the interval #[0,\infty)# so
[tex]L_n(x)=_1F_1(-1;n;x)[/tex]
What that means? That degenerate hypergeometric function converge in the interval #[0,\infty)#?
Hermitte polynomials are defined in the interval #(-\infty,\infty)#.
[tex]H_n(x)=(2x)^n_2F_0(\frac{1-n}{2},-\frac{n}{2};\_;-\frac{1}{x^2})[/tex]
Is there some easy way to see in what interval [tex]_p F_q[/tex] converge?
 
  • #11
matematikuvol said:
How you get connection between Legendre polynomial and #_2F_1#?
Sorry, it should be too long and boring to copy on the forum the developments which were done a long time ago, leading to this result. The relationships between the Gauss hypergeometric functions and various functions of lower level, Jacoby polynomials, Legendre polynomials, etc. can be found in many handbooks of special functions ( in attachment)
 

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1. What is a Hypergeometric function?

A Hypergeometric function is a mathematical function that arises in the field of mathematics known as special functions. It is defined as a solution to the hypergeometric differential equation and is used to describe a wide range of phenomena in mathematics, physics, and engineering.

2. How is a Hypergeometric function different from other special functions?

Unlike other special functions, the Hypergeometric function can be expressed in terms of elementary functions, making it easier to manipulate and use in various mathematical calculations. It also has a wide range of applications, making it a versatile tool in many scientific fields.

3. What are the key properties of a Hypergeometric function?

The key properties of a Hypergeometric function include its convergence behavior, its transformation properties under various operations, and its relationship with other special functions. It also has specific summation formulas and identities that are useful in solving various mathematical problems.

4. How is a Hypergeometric function used in statistics?

In statistics, the Hypergeometric function is used to describe the probability of obtaining a specific combination of outcomes in a sample when the population is finite. It is commonly used in experiments where the sample size is relatively small compared to the population size, such as in quality control and market research.

5. Can the Hypergeometric function be extended to higher dimensions?

Yes, the Hypergeometric function can be extended to higher dimensions, known as the multivariate Hypergeometric function. It is used to describe the probability of obtaining a specific combination of outcomes in multiple samples when the population is finite. It has applications in fields such as genetics, ecology, and epidemiology.

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