Hypergeometic function. Questions.

[LagrangeEuler]

Legendre polynomial is defined as
$P_n(x)=_2F_1(-n,n+1;1,\frac{1-x}{2})$
Pochammer symbols are defined as $(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}$. If I undestand well
$$P_n(x)=_2F_1(-n,n+1;1,\frac{1-x}{2}) =\sum^{\infty}_{k=0}\frac{(-n)_k(n+1)_k}{(1)_kk!}x^k$$
I am not sure what happens with
$$(-n)_k=\frac{\Gamma(-n+k)}{\Gamma(-n)}$$
because $\Gamma$ diverge for negative integers.

 

[Geofleur]

Even though $\Gamma$ diverges for negative integers, the ratio of gamma functions evaluated at different integer arguments does not necessarily diverge. First, note that $\Gamma(n) = (n-1)!$. Now write out the terms as follows:

$\frac{\Gamma(-n+k)}{\Gamma(-n)} = \frac{(k-n-1)!}{(-n-1)!} = \frac{(k-n-1)(k-n-2)\cdots (k-n-(k-1))(k-n-k)(k-n-(k+1))\cdots }{(-n-1)(-n-2)(-n-3)\cdots}.$

Therefore, we have

$\frac{\Gamma(-n+k)}{\Gamma(-n)} = \frac{(k-n-1)(k-n-2)\cdots (-n+1)(-n)(-n-1)\cdots }{(-n-1)(-n-2)(-n-3)\cdots} = (k-n-1)(k-n-2)\cdots(-n+1)(-n)$,

and this last result is finite.

 

[math771]

Can you clarify this for me?

Sure, I can try to clarify this for you. The Pochhammer symbol, also known as the rising factorial, is defined as (a)_n = a(a+1)(a+2)...(a+n-1). This definition holds for any real or complex number a, including negative integers. However, as you mentioned, the gamma function, which is used in the definition of the Pochhammer symbol, diverges for negative integers. This is because the gamma function is only defined for positive real numbers.

So, how do we deal with the Pochhammer symbol for negative integers? We use the reflection formula for the gamma function, which states that Γ(z)Γ(1-z) = π/sin(πz). This allows us to rewrite the Pochhammer symbol as (a)_n = Γ(a+n)/Γ(a) = Γ(a+n)Γ(1-a)/Γ(a) = π/(sin(πa)sin(π(1-a))...sin(π(1-a-n+1)).

Now, when we substitute this into the Legendre polynomial formula, we get P_n(x) = ∑(k=0 to ∞) (a)_k x^k = π/sin(πa)∑(k=0 to ∞)x^k sin(π(1-a-k)). This infinite sum can be simplified using the geometric series formula, and we end up with the final form of the Legendre polynomial: P_n(x) = π/(sin(πa)√(1-x^2)).

So, in summary, even though the gamma function diverges for negative integers, we can still use the reflection formula to define the Pochhammer symbol for negative integers and use it in the Legendre polynomial formula. I hope this helps clarify things for you!