I'm investigating a series that intertwines Bessel functions, partitions, and multinomial coefficients, aiming to derive a closed-form expression or deepen understanding of its properties. The series in question is: $$ \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} J_0\left(\pi\sum_{\substack{k=1 \\ r_1 \geq 0, \ldots, r_K \geq 0 \\ r_1 + \ldots + r_K = n}}^K k \cdot r_k\right) (x_1 + \ldots + x_K)^n $$ for determined $K$.
Context: The expression combines a Bessel function $J_0$ with a sum over integer partitions and a multinomial expansion. I'm exploring using the Taylor expansion of $e^{-z}$ and properties of the product of Fourier series coefficients to simplify or understand this series better.
Questions:
How can the Taylor expansion of $e^{-z}$ be applied here, especially given the exponential and Bessel function interaction? Could the product of Fourier series coefficients provide insights into the sum's behavior, considering the series' combinatorial nature?
Attempts: I've looked into isolating components and examining their properties but am struggling with integrating these into a coherent approach.
Seeking: Any insights, references, or strategies that could aid in understanding or simplifying this complex series.
Thank you for any guidance!
Multinomial Expansion Definition
The multinomial expansion refers to the expansion of a power of a sum that includes more than two terms. For a given expression of the form ($(x_1 + x_2 + \cdots + x_K)^n$), the multinomial expansion can be expressed as:
$$ (x_1 + x_2 + \cdots + x_K)^n = \sum \binom{n}{n_1, n_2, \ldots, n_K} x_1^{n_1} x_2^{n_2} \cdots x_K^{n_K} $$