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I'm trying to isolate $x$ from the next equation. Some ideas?

\begin{equation} k_{1} = k_{2}\sum_{i=1}^{x}\binom{x-1}{i-1}\frac{1}{i^{k_{3}}}, \end{equation}

where $k_{1}$, $k_{2}$ and $k_{3}$ are constants. If not possible to do it exactly, an approximation can also be used.

Thank you in advance by your help!

Henry
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  • You are unlikely to get a closed form except in the simplest cases. If you have a less simple case I would use software. – almagest Jun 25 '16 at 20:38
  • Thank you @almagest. The simplest case for my problem is when k3 is a positive integer. greater than 2. – Henry Jun 25 '16 at 20:43

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Forgive me as this is not entirely an answer (I do not have comment privileges) but by observing only $$f(c)=\sum_{i=1}^x\binom{x-1}{i-1} \dfrac{1}{i^c}:c\in\mathbb{Z}$$

we can make some observations. Of course, in your case, $k_3\in\mathbb{R}$

I have listed some observations with various integer values below:

\begin{align} &\quad\!\!\!\cdot\\ &\quad\!\!\!\cdot\\ &\quad\!\!\!\cdot\\ f(-3)&=(x+1)(x^2+5x-2)2^{x-4}\\ f(-2)&=x(x+3)2^{x-3}\\ f(-1)&=(x+1)2^{x-2}\\ f(0)&=2^{x-1}\\ f(1)&=\frac{2^{x-1}}{x}\\ f(2)&=_3\!\!F_2(1,1,1-x;2,2;-1)\\ f(3)&=_4\!\!F_3(1,1,1,1-x;2,2,2;-1)\\ &\quad\!\!\!\cdot\\ &\quad\!\!\!\cdot\\ &\quad\!\!\!\cdot\\ \end{align}

where $_pF_q(a_1,\cdots,a_p;b_1,\cdots,b_q;x)$ is the generalized hypergeometric function given by $$_pF_q(a_1,\cdots,a_p;b_1,\cdots,b_q;x)=\sum_{k=0}^\infty \dfrac{(a_1)_k\cdots(a_p)_kx^k}{(b_1)_k\cdots(b_q)_kk!}$$

We do notice some behavior with the function $f(c)$ as the polynomial "coefficient" decreases its degree by 1 with on increase of $c$ by $1$ and the exponent power increases by $1$ up until $f(0)$.

In general for $c=k_3\in\mathbb{Z^{>2}}$,

$$f(c)=_{c+1}F_{c}(\underbrace{1,\cdots,1}_{c+1 \quad\!\!\! \text{times}},1-x;\underbrace{2,\cdots 2}_{c \quad\!\!\! \text{times}};-1)$$

Jake
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  • Thank you by your help Hannah! – Henry Jun 25 '16 at 20:10
  • Looking at your comment reply to almagest, I believe your best bet is by looking at the generalized hypergeometric function. I will edit my post to adjust my answer to your needs. – Jake Jun 25 '16 at 20:45
  • Thanks Hannah, you are great! Is it possible to isolate x from f(c)? For now I have no privileges to give +1 to your answer, sorry for that! – Henry Jun 25 '16 at 20:53
  • A closed-form is highly unlikely. Sorry! – Jake Jun 25 '16 at 20:56