Stumpff function

In celestial mechanics, the Stumpff functions ck(x), developed by Karl Stumpff, are used for analyzing orbits using the universal variable formulation. They are defined by the formula: $$c_k (x) = \frac{1}{k!} - \frac{x}{(k + 2)!} + \frac{x^2}{(k + 4)!} - \cdots = \sum_{n=0}^\infty {\frac{(-1)^n x^n}{(k + 2n)!}}$$ for $$k = 0, 1, 2, 3,\ldots$$ The series above converges absolutely for all real x.

By comparing the Taylor series expansion of the trigonometric functions sin and cos with c0(x) and c1(x), a relationship can be found: $$\begin{align} c_0(x) &= \cos {\sqrt x}, \\[1ex] c_1(x) &= \frac{\sin {\sqrt x}}{\sqrt x}, \end{align} \quad \text{ for }x > 0$$ Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find: $$ \begin{align} c_0(x) &= \cosh {\sqrt {-x}}, \\[1ex] c_1(x) &= \frac{\sinh {\sqrt {-x}}}{\sqrt {-x}}, \end{align} \quad \text{ for }x < 0$$

The Stumpff functions satisfy the recurrence relation: $$x c_{k+2}(x) = \frac{1}{k!} - c_k(x),\text{ for }k = 0, 1, 2, \ldots\,.$$

The Stumpff functions can be expressed in terms of the Mittag-Leffler function:

$$c_{k}(x) = E_{2,k+1}(-x).$$