The factorials and the binomial functions are connected with combinatorial problems from a long period of time. The combinatorial problems are introduced since the ancient times and were partially solved using rigorous methods. But ever since introduction of the factorials expressing the binomial coefficients of were described using combinatorial expressions and thus becomes very simple expression. The mathematicians Yang Hui, Tshu shi Kih, Shih–Chieh Chu and others found a recursion relation with the binomial relation and the fractional arguments in factorials are studied by I. Newton. The growth of n! is very fast as becomes larger and the growth rate was estimated by a asymptotic formula given by Scottish mathematician James Stirling. This is known as Stirling’s formula
The factorial plays an important role for determining the value of Gamma function. Euler approximated the gamma function in terms of factorial numbers (Γ(n) = (n-1)!). The factorial is also used in non-integer arguments with the notations and the parentheses for the binomials. Hindenburg used the factorials to generalize the multinomial expressions. The modern factorial symbol (n!) was first introduced by C. Kramp and the binomial symbol was first introduced by A. von Ettinghausen. The operator factorial n!! was introduced far more years ago but its application in the complex arguments was first suggested few years back in 1994 by J. Keiper and O.I. Marichev when implementing Factorial2 in the programming language Mathematica.
By the definition of n! the ways by which n things can be arranged in n places is the quantity given by n! = n*(n-1)*(n-2)…1. For example, 5! = 5*4*3*2*1 = 120. It should be noted that 0! = 1 as the all possible combination of choosing nothing from set of nothing is 1.
The binomial symbol is the all possible selections of k things out of n things (where n>= k). The multinomial (n;n1,n2,…,nm) is the all possible ways to put n = n1 + n2 + n3+…+ nm different numbers of objects inside m different boxes where nk in the kth box number, where, k = 1,2,…,m.
The terms of different factorials are given by the following formulas.
n! = Γ(n+1)
Where, n!! = n double factorial.
Where, = Pochhammer symbol.
An alternative form of Pochhammer symbol is
Multinomial expression is given by,
When α = a and v = n are integers having values a<=0 and n <= -a, the Pochhammer symbol can’t be defined in a unique way. This is represented by limits as the two variables α and approaches the integers a and n respectively in different speeds. Now, for the integers a <= 0, n <= -a the formula which is used is
The binomials and Pochhammer functions has no singularities or there are no points where the function is not defined in case of positive arguments. However, in case of negative arguments both of the functions have many singularities.
Properties of factorials and binomials:
For any real value of the arguments values of factorials and binomials like n!, n!!, (a)_n, and (n1+n2+…+nm;n1,n2,...,nm) have the real or infinite value.
If the arguments are zero then the values of factorials and binomials are simple and finite. In other words
In case the variable n is an integer or a rational number then n! and n!! can be presented by the general formulas given by,
Some particular value of the Pochhammer symbol are given below.
Some of the formulas of multinomial and binomials are
Factorials and binomials for complex variables:
The binomials and their factorials for their complex values are valid in n!, n!!, (a)_n and . However, the factorials and binomials of complex variables do not have branch points and branch cuts. The n! and double factorial n!! do not have zero values and hence (1/n!) and (1/n!!) do not have singularities at zero but only singular point is z =
Poles and the essential singularities:
The factorial n! has infinite amount of singular points at n = -k; where k ∈ N+ have the poles with the residues The double factorial n!! has infinite amount of singular points at n = -2k ; where k -1 ∈ N+ are the poles having the residues For a specific a the function has infinite number of points of singularity at n = -a –k; k ∈ N having the residues
When n is specific then the function has infinite set of singularities at a = -k – n ; where k ∈ N are the poles having the residues (-1)^k /(k! Γ(-n-k)) where k + n ∉ N.
When k is specific then the function has an infinite number of singular points at n = -j; j ∈ N+ having the residues , where k ∉ Z.
When k in [1,m] where m is any finite positive integer not equal to zero, the function
(n1 + n2 +… + ; n1, n2, n3,…) has infinite amount of singularities at = and the residues of the function are
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