Has anyone seen the definition of "alpha function" from Wolfram MathWorld in cited references?

Question

This entry in WolframMathWorld define the "alpha function" as $$ \begin{aligned} \alpha_{n}(z) & \equiv \int_{1}^{\infty} t^{n} e^{-z t} d t =n ! z^{-(n+1)} e^{-z} \sum_{k=0}^{n} \frac{z^{k}}{k !} \end{aligned} $$

Unlike other ubiquitous special functions such as beta function and gamma function, I have never seen it in any book I read before. I suspect that if this is simply an entry made up by Eric Weisstein, who is the creator of MathWorld.

Euler certainly had no such notion as this post (Did Euler have an alpha function?) on the site shows. A search on Google does not return anything related to the definition above.

Has anyone seen this definition of "alpha function" in books or journal articles?

Answer 1

See these:

• On some properties of the α-Exponential function Jigen Peng &Jinghuai Gao
• The k-α-Exponential Function Luciano L. Luque and Ruben A. Cerutti

Answer 2

The alpha function is a more uncommon name for the Generalized Exponential Integral which itself is just an Incomplete Gamma function.

Notice that from our bolded link:

$$\text E_v(z)=\Gamma(1-v) z^{v-1}-\sum_{k=0}^\infty \frac{(-k)^n}{(k-v+1)k!}$$

Meanwhile:

$$\alpha_n(z)=n! z^{-n-1}e^{-z} \sum_{k=0}^n \frac{z^k}{k!} $$

and the Regularized Gamma function gives:

$$Q(n+1,z)= \frac{\Gamma(n+1,z)}{\Gamma(n+1)} =e^{-z} \sum_{k=0}^{n}\frac{z^k}{k!}, n\in\Bbb N$$

Therefore:

$$\alpha_n(z)=n! z^{-n-1}e^{-z} \sum_{k=0}^n \frac{z^k}{k!}= n!z^{-n-1}\frac{\Gamma(n+1,z)}{n!}=\frac{\Gamma(n+1,z)}{z^{n+1}}$$

is just an elementary function times an Incomplete Gamma function which has hundreds of papers behind it.

We can also use the following relation between the Generalized Exponential Integral function and Incomplete Gamma function:

$$\frac{\Gamma(a,z)}{z^a}=\text E_{1-a}(z)$$

Therefore:

$$\alpha_n(z)=\text E_{-n}(z)$$

so you do not really need the alpha function, just the Incomplete Gamma function or Genralized Exponential Integral which also has hundreds of papers about it both of which are much more commonly used. Please correct me and give me feedback!