New Concept of Factorials and Combinatorial Numbers and its Consequences for Algebra and Analysis

: In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has been proposed, which led directly to the non - classical hypergeometric orthogonal polynomials and the non - classical second - order hypergeometric linear DEs. The resulting polynomials can be used in non - relativistic and relativistic QM, particularly, in the case of the Schrödinger equation, and Dirac equations for an electron in a Coulomb potential field.


Introduction
Usually, the factorial of a positive integer n , denoted by ! n , is defined as the product of all positive integers less than or equal to n: . The value of 0! is conventionally equal to 1. However, the factorials of negative integers cannot be computed, since for n = 0, the recurrence relation n n n / ! )! 1 (   implies division by zero, and also due to the fact that the usual binomial coefficient is always equal to a positive integer. The factorials are mostly encountered in many areas of mathematics, notably in combinatorics, probability theory, number theory, statistics, algebra, and mathematical analysis. Actually, their most basic use counts the possible distinct sequences the 'permutations' of n distinct objects: there are n!. The factorials can also be extended to real numbers while retaining its most important properties. This involves using gamma function to define ! ) . But, as it is abovementioned, this extension does not work when x is a negative integer.
The factorials have a long and fascinating history [1,2]. They were used to count permutations at least as early as the 12th century, by Indian scholars [3]. In 1677, Fabian Stedman described factorials as applied to change ringing, a musical art involving the ringing of many tuned bells [4]. In 1808, the French mathematician Christian Kramp introduced the notation n! [5].
Concerning the gamma function and the extension of factorials to real negative numbers, many famous mathematicians worked on this topic, particularly Euler, Bernoulli (Daniel) Goldbach and De Moivre [6,7].
As a matter of fact, the field of factorials has attracted many modern researchers whose goal was the generalization of factorials and/or the extension of gamma function [8,9,10,11,12]. However, the purpose and expectations of the present work are radically different from what was already published on the topic under discussion as we will see soon.

Generalization of factorials to negative integers
The usual factorials of positive integers can be generalized to negative integers as follows. We have It is quite clear that multiplying each integer on the RHS of (1) by the constant 1   ε is equivalent to multiply the LHS of (1) by n ε) ( : (3) Using the notation ) to rewrite (3) in the following compact form As we can see, the relation (4) is, in fact, a special case of We call the relation (4) 'factorials of negative integers', therefore, Eq.(5) may be understood as a generalization of factorials to negative integers and beyond. Now, return to (1) and rewrite it in the following form , Multiplying the two sides of (6) by 1  to get From (4) and (7), we arrive at the result: in general Definition 1: we call a factorial number any expression of the form (4) and (5) we get where N Z    n and    Z n .

Generalization of binomial coefficients to negative integers
The previous generalization of factorials to negative integers allows us to generalize the binomial coefficients to negative integers along these lines. We have Replacing n by n  in both sides of (10) to get Taking into account Eq.(4) and definition (1), the expression (11) becomes Again, replacing k by k  in both sides of (10) yields Result 2: from (12) and (13), we get Finally, replacing n and k by n  and k  , respectively, in both sides of (10) to obtain Result 3: comparing (9) and (15) yields Definition 2: we call a combinatorial number any expression of the form (12), (13) and (15).
As we will see, the usual factorials of positive integers are, actually, a special case of Eq. (5), that is to say, when 1   . Furthermore, the usual binomial coefficients are generalized to negative integers via the formulae (12), (13) and (15). To clarify all that, some usual factorials of positive integers, factorial numbers (factorials of negative integers), usual binomial coefficients and combinatorial numbers are, respectively, listed in Tables 1 and 2.

Some generalized combinatorial formulae
We deduce from definitions 1 and 2 the following interesting formulae which can be useful later.

Application of the generalized combinatorial numbers
The concept of the generalized combinatorial numbers as a generalization of the usual binomial coefficients allows us to introduce some new kind of polynomials in which the generalized combinatorial numbers defined by formulae 12, 13, 15 and I-VI, playing the role of coefficients. At present, let us begin with the polynomial    where  C {12, 13, 15, I, II, III, IV, V, VI} is the set of the generalized binomial coefficients defined by the formulae 12, 13, 15 and I-VI. Furthermore, in order to understand correctly the role and importance of k n, a we make use of (17) to define the following polynomials: First, we begin with (18) which can be written in explicit form as follows : The first few P-polynomials

Property
Now, supposing y is fixed thus by putting we get, after substitution in (20), the polynomial   α , z P n of order α and degree n in z : The polynomial (26) satisfies the second-order self-adjoint DE   for the interesting special case when (28)

Non-classical hypergeometric orthogonal polynomials
The well-known sets of orthogonal polynomials (OPs) of Jacobi polynomials 0 n a n x y (including as an important special case the Legendre polynomials, Chebyshev polynomials, and Gegenbauer polynomials) are usually called classical (hypergeometric) OPs [16,17,18,19] because they satisfy the well-known four classical hypergeometric linear DEs, namely: Jacobi Eq.: Hermite Eq.: Also the classical OPs have the property that their derivatives form orthogonal systems.
In addition to the above mentioned classical OPs, there is a new class of OPs do not belong to the classical sets of OPs and they are called non-classical (hypergeometric) OPs for the reason that besides their property of orthogonality, they satisfy a non-classical second-order hypergeometric linear DE of the form: such that the coefficients of u   and u are quadratic and/or linear polynomials; ) , ( β α and n are, respectively, bi-order and degree of the polynomial solutions of Eq.(29); the characteristic parameter n λ depends on the polynomial's degree 'n' and is defined by The weight function of polynomial solutions is given by and satisfies the condition where a and b are the end points of the interval of orthogonality of polynomial solutions and As we can see, Eq.(29) is different from the second-order classical hypergeometric DEs (i-iv) in that-the coefficient of u contains the positive integral parameter 'n' which is the degree of polynomial

4.1.-polynomials
Throughout the rest of this paper we concentrate exclusively on the derivation of -polynomials and their DE, generating function, recurrence relation, derivative formula, weight function, Rodrigues' formula, characteristic values as well as their properties . But first, let us begin with their derivation as follows. Assuming y is fixed thus by letting we get, after substitution in (19): are the very expected -polynomials of bi-order ) , ( β α and degree n in z , which satisfy the nonclassical second-order hypergeometric linear DE: Furthermore, the -polynomials defined by the following explicit formula are also solutions of Eq.(36).
The first few -polynomials Recurrence relation Derivative formula The analogue of Rodrigues' formula

Properties of -polynomials
Characteristic values

-monomial
We get the -monomial for the interesting special case when

Orthogonality
Now we prove the orthogonality of -polynomials w.r.t the weight function (42) on the interval (0, ). That is to say, we will get n m  . To this end, let us rewrite Eq.(36) in the following self-adjoint form Then by Eq. (45) we get (47) Integrating the first integral by parts we get In exactly the same way we can multiply (47) by (49) Finally, subtracting (49) from (48) we get the very expected orthogonality condition The first term on the RHS is zero, hence by integrating several times by parts, we find Finally, combining (50) and (56) we get and mn δ is the Kronecker delta defined as

Series of -polynomials
As a direct consequence of -polynomials we can refer to the series of -polynomials, that is to say, any continuous function ) (z f on the interval ) , 0 (  such that 1  z , may be expanded in series of polynomials. More precisely, let us prove that if For the case when n m  , we have from where we can obtain the very expected formula (59).

Consequences of -polynomials
The consequences and applications of -polynomials are generally related to the specialization of their bi-order (,). For instance, as a direct consequence of -polynomials we can refer to the series of polynomials. It may be added that the -polynomials and their reverse can be used in physics particularly in non-relativistic and relativistic quantum mechanics as we will see. With this aim, we shall now consider the following interesting special case, that is, when where, as usual, and N  n . Therefore, after substitution, the -polynomials (35) and their DE (36) become The first few -polynomials for It is worthwhile to note that in spite of the fact that Eq.(63) is identical to the so-called Laguerre (generalized) equation (2) , the polynomials (62) are not strictly speaking identical to the alleged generalized (associated) Laguerre polynomials       z z z z n a z n n a n n a n n a n z n a n n a n n a n d d L n a n a n a n This implies, among other things, that they have not the same generating function. Consequently, the polynomials   (65) The first six  -polynomials As we can see from (64) or equivalently   z z L n a n n a n Expressions (68) and (69) Because of this, the polynomials   z n a n ) , (    can be arisen in the treatment of the quantum harmonic oscillator.

Associated φ-functions
In terms of the polynomials   Using Eq.(63), we get the following self-adjoint DE for φ-functions Therefore, φ-functions are eigen-functions of a Sturm-Liouville system on the interval (0, ).

-polynomials
which are also solutions of Eq.(63).
The first six -polynomials The analogue of Ridrogues' formula   z n a n n z a n n a n e z dz d e z n a z The polynomials (74) are related to   z n a n ) , (    for  =a and  =n by the following relation: z n n a z n a n n n n a n π π ) , ( ) ( ) , It may be added that the orthogonality property of the polynomials (74) is a direct consequence of the orthogonality of -polynomials (35) and without difficulty can be shown to be of the form:

-monomials and -polynomials
In addition to the properties of -polynomials (35) mentioned in the previous section, there is another specific property which is in fact a direct consequence of the explicit expression of -polynomials which can also be written in the form of the product of -monomials in z and -polynomials in (z -1 ) as follows:

Orthogonality
To show the orthogonality property of -polynomials on the interval (0, ) w.r.t the weight function (86) it suffices to follow the usual procedure. With this aim, let us first rewrite Eq.(81) in self-adjoint form.