hypergeometric function with special Pochhammer symbol

Question

What is the relationship between these two hypergeometric functions? Can the following function be written as another function of some hypergeometric functions ? $$1F1(a+b,2a,x)$$ and $$1F1(a+b,a,x)$$ Can I convert 'a' to '2a'?

Answer 1

We have that $$ {}_1F_1 (a + b,2a,x) = \sum\limits_{0\, \le \,k} {{{\left( {a + b} \right)^{\,\overline {\,k\,} } } \over {\left( {2a} \right)^{\,\overline {\,k\,} } }}{{x^{\,k} } \over {k!}}} $$

where the exponent overlined is a way to indicate the Rising Factorial or Pochammer symbol.

Now, the denominator equals $$ \eqalign{ & \left( {2a} \right)^{\,\overline {\,k\,} } = \prod\limits_{0\, \le \,j\, \le \,k - 1} {\left( {2a + j} \right)} = \cr & = \prod\limits_{0\, \le \,2j\, \le \,\left\lfloor {{{k - 1} \over 2}} \right\rfloor = \left\lceil {{k \over 2}} \right\rceil - 1} {\left( {2a + 2j} \right)} \prod\limits_{0\, \le \,2j\, \le \,\left\lfloor {{k \over 2}} \right\rfloor - 1} {\left( {2a + 1 + 2j} \right)} = \cr & = 2^{\,k} \prod\limits_{0\, \le \,2j\, \le \,\left\lfloor {{{k - 1} \over 2}} \right\rfloor = \left\lceil {{k \over 2}} \right\rceil - 1} {\left( {a + j} \right)} \prod\limits_{0\, \le \,2j\, \le \,\left\lfloor {{k \over 2}} \right\rfloor - 1} {\left( {a + 1/2 + j} \right)} = \cr & = 2^{\,k} a^{\,\overline {\,\left\lceil {{k \over 2}} \right\rceil \,} } \left( {a + 1/2} \right)^{\,\overline {\,\left\lfloor {{k \over 2}} \right\rfloor \,} } \cr} $$

or, alternatively $$ \left( {2a} \right)^{\,\overline {\,k\,} } = \left( {2a} \right)^{\,\overline {\,k + a - a\,} } = \left( {2a} \right)^{\,\overline {\, - a\,} } \left( a \right)^{\,\overline {\,k + a\,} } $$

In both versions it is not possible to simply relate the above results with $a^{\,\overline {\,k \,} }$.