Understanding Notation and Connections of Hypergeometric Functions

[matematikuvol]

Hypergeometric function is defined by:
_2F_1(a,b,c,x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n
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
_2F_1(-n,b,b,1-x)=\sum^{\infty}_{n=0}\frac{(-n)_n}{n!}(1-x)^n
or
_2F_1(-n,b,b,1-x)=\sum^{\infty}_{k=0}\frac{(-n)_k}{k!}(1-x)^k
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?

 

[JJacquelin]

Hi !

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

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

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

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 :
_2F_1(-n,b,b,1-x)=x^n

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

 

[matematikuvol]

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?

 

[dextercioby]

Check out the connection between the two. It involves a limit on one of the parameters.

 

[JJacquelin]

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

 

[matematikuvol]

Tnx a lot for usefull answer. All hypergeometric function converge for $|x|<1$. Right? So for example
\ln (1+x)=x_2F_1(1,1;2;-x)
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
\frac{1}{\sqrt{1-2xt+t^2}}=\sum^{\infty}_{k=0}P_k(x)t^k
What is easiest way to get relationship beetween $P_n$ and some of the family of hypergeometric functions.

 

[JJacquelin]

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.

 

[LagrangeEuler]

Tnx a lot for usefull answer. All hypergeometric function converge for $|x|<1$. Right? So for example
\ln (1+x)=x_2F_1(1,1;2;-x)
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
\frac{1}{\sqrt{1-2xt+t^2}}=\sum^{\infty}_{k=0}P_k(x)t^k
What is easiest way to get relationship beetween $P_n$ and some of the family of hypergeometric functions.

No. Kummer function converge for all $x$.

 

[matematikuvol]

?

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#?

 

[matematikuvol]

#_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
P_n(x)=_2F_1(-n,n+1;1;\frac{1-x}{2})
From this table I see that
T_n(x)=_2F_1(-n,n;\frac{1}{2};\frac{1-x}{2})
Laquerre polynomials are defined on the interval #[0,\infty)# so
L_n(x)=_1F_1(-1;n;x)
What that means? That degenerate hypergeometric function converge in the interval #[0,\infty)#?
Hermitte polynomials are defined in the interval #(-\infty,\infty)#.
H_n(x)=(2x)^n_2F_0(\frac{1-n}{2},-\frac{n}{2};\_;-\frac{1}{x^2})
Is there some easy way to see in what interval _p F_q converge?

 

[JJacquelin]

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)