Yes, it is possible to express the reciprocal Gamma function, $\frac{1}{\Gamma(z)}$, using a series involving trigonometric functions. One way to achieve this is by using the Weierstrass product form of the Gamma function and then expressing it in terms of trigonometric functions.

1. Weierstrass Product Representation of the Gamma Function

The Gamma function $\Gamma(z)$ can be represented by the Weierstrass product formula as follows:

$\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right)^{-1} e^{z/n},$

where $\gamma$ is the Euler-Mascheroni constant. Taking the reciprocal, we get:

$\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-z/n}.$

2. Expression Using Sine Function

Using a trigonometric representation, the reciprocal Gamma function can be linked to the sine function through the reflection formula of the Gamma function:

$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}.$

Taking the reciprocal of the Gamma function, we get:

$\frac{1}{\Gamma(z)} = \frac{\sin(\pi z)}{\pi \Gamma(1-z)}.$

This relation allows us to express $\frac{1}{\Gamma(z)}$ in terms of sine and the Gamma function evaluated at $1-z$. However, expressing $\frac{1}{\Gamma(z)}$ purely in terms of trigonometric functions without involving the Gamma function of another argument directly can be more complex.

3. Series Expansion Approach

The reciprocal Gamma function can also be expanded as a Laurent series around $z = 0$ and other points, using series involving trigonometric functions like sine and cosine indirectly through various expansions of the Gamma function.

One notable series that approximates the reciprocal Gamma function using trigonometric functions is:

$\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 - \frac{z}{n} + \frac{z^2}{2n^2} - \cdots \right).$

By manipulating these expansions, terms involving $\sin(\pi z)$ and other trigonometric functions may emerge.

Conclusion

While there isn't a simple, purely trigonometric series that directly gives $\frac{1}{\Gamma(z)}$, using the reflection formula and Weierstrass product forms provides a connection between $\frac{1}{\Gamma(z)}$ and trigonometric functions. These relationships highlight the deep interplay between the Gamma function and trigonometric functions in complex analysis.

Would you like to explore more details or have any specific questions?

Additional Questions to Explore:

1. How can the Weierstrass product be used to derive other properties of the Gamma function?
2. What are the implications of the reflection formula on the Gamma function's behavior?
3. How can we derive the Euler-Mascheroni constant from the properties of the Gamma function?
4. What is the significance of the Laurent series expansion of the reciprocal Gamma function?
5. How does the digamma function relate to the series representation of $\frac{1}{\Gamma(z)}$?

Tip:

When dealing with the Gamma function and its properties, always consider its connections to factorials, integrals, and trigonometric functions, as these relationships often provide deeper insights into the function's behavior and applications.