What Is the Arithmetic Gamma Function \(\gamma_{m}(n)\)?

[elibj123]

In some exercises I've stumbled upon a function which is denoted $\gamma_{m}(n)$ with m,n natural. I've no idea what is the definition of the function and could not infer from the exercises. Searching google yielded nothing, as it kept suggesting me the OTHER Gamma function.
Can anyone here help me please?

Thanks in advance

 

[mathman]

There is something called the incomplete gamma function which uses the symbol (lower case gamma) that you have, but it does look a little different.

 

[Stephen Tashi]

elibj123,

Post some of the exercises. Perhaps someone will recognize it from the context.

 

[Hurkyl]

elibj123,

Post some of the exercises. Perhaps someone will recognize it from the context.

And what he is actually studying! (if not clear from the exercise)

 

[blue_raver22]

for any help!

The arithmetic gamma function, denoted as \gamma_{m}(n), is a mathematical function that is defined for natural numbers m and n. It is different from the commonly known gamma function, which is denoted as \Gamma(x) and is defined for all complex numbers x except for negative integers. The arithmetic gamma function has been used in some exercises, but its definition is not well-known and cannot be easily inferred.

To understand the arithmetic gamma function, we need to look at its properties and possible applications. One possible definition of the function is as follows: \gamma_{m}(n) = \frac{n!}{(n-m)!} = n(n-1)(n-2)...(n-m+1). This definition is similar to the factorial function, but it only considers the first m terms instead of all natural numbers less than or equal to n. This function can be useful in combinatorics and probability, where m represents the number of objects chosen from a set of n objects.

However, this is just one possible definition of the arithmetic gamma function and there may be other ways to define it. Its properties and applications may also vary depending on the specific definition used. Therefore, it is important to clarify the exact definition when using this function in exercises or research.

In conclusion, the arithmetic gamma function is a mathematical function that is defined for natural numbers m and n, and it is different from the commonly known gamma function. Its exact definition may vary and it may have various applications in mathematics. Further research and clarification may be needed to fully understand and utilize this function.