The κ-Generalizations of Stirling Approximation and Multinominal Coefficients

As is well-known in the fields of statistical mechanics, Ludwig Boltzmann clarified the concept of entropy. He considered that a macroscopic system consists of a large number of particles. Each particle is assumed to be in one of the energy levels, Ei (i = 1, . . . , k), and the number of particles in the energy level, Ei, is denoted by ni. The total number of particles is then n = ∑k i=1 ni, and the total energy of the macrostate is E = ∑


Introduction
As is well-known in the fields of statistical mechanics, Ludwig Boltzmann clarified the concept of entropy.He considered that a macroscopic system consists of a large number of particles.Each particle is assumed to be in one of the energy levels, E i (i = 1, . . ., k), and the number of particles in the energy level, E i , is denoted by n i .The total number of particles is then n = k i=1 n i , and the total energy of the macrostate is E = i n i E i .The number of ways of arranging n particles in k energy levels, such that each level, E i , has n i , (i = 1, 2, . . ., k) particles, are given by the multinominal coefficients: which is proportional to the probability of the macrostates if all microscopic configurations are assumed to be equally likely.In the thermodynamic limit, in which n increases to infinity, we consider the relative number, p i ≡ n i /n, as the probability of the particle occupation of a certain energy level, E i .In order to find the most probable macrostate, Equation ( 1) is maximized under a certain constraint.The Stirling approximation of the factorials leads to: where is the Boltzmann-Gibbs-Shannon (BGS) entropy.
Recently, one of the authors (H.S.) has shown one-parameter (q) generalizations of Gauss' law of error [1], Stirling approximation [2] and multinominal coefficients [3] based on the Tsallis q-deformed functions and associated multiplication operation (q-product) [4,5].These mathematical structures are quite fundamental for the basis of any generalization of statistical physics as in the standard statistical physics.In particular, [3] has shown the one-to-one correspondence between the q-multinominal coefficient and Tsallis q-entropy [6-8], i.e., the one-parameter (q) generalization of Equation (2).A combinatorial form of Tsallis entropy was derived in [9].
On the other hand, Kaniadakis [10][11][12] has proposed the κ-generalized statistical mechanics based on the κ-deformed functions, which are the different type of one-parameter deformations for the exponential and logarithmic functions.Based on the κ-deformed functions and the associated product operation (κ-product), we have already shown the κ-generalization of Gauss' law of error [13].
In this work, we show the κ-generalizations of the Stirling approximation and the relation between the κ-multinominal coefficient and κ-entropy.In the next section, the κ-factorial is introduced based on the κ-product.Then, the κ-generalization of the Stirling approximation is obtained.We see the failure of the naive approach relating the κ-multinominal coefficients with the κ-entropy.In order to overcome this difficulty, we introduce a new kind of κ-product in Section 3 and show the explicit relation between the κ-multinominal coefficients and the κ-entropy.As a complemental generalization, we explain another type of the κ-factorial, which is based on the κ-generalization of gamma function in Section 4. The final section is devoted to our conclusions.

κ-Stirling Approximation
Let us begin with the brief review of the κ-factorial and its Stirling approximation.The κ-generalized statistics [10][11][12] is based on the κ-entropy: where κ is a real parameter in the range (−1, 1), and the κ-logarithmic function is defined by: Its inverse function is the κ-exponential function defined by: In the limit of κ → 0, both the κ-logarithmic and κ-exponential functions reduce to the standard logarithmic and exponential functions, respectively.We thus see that S κ reduces to the BGS entropy of Equation ( 3) in the limit of κ → 0.
Based on the above κ-deformed functions, the κ-product is defined by: which reduces to the standard product, x • y, in the limit of κ → 0. From the definition, we readily see that this κ-product satisfies the associative and commutative laws [10][11][12].
Similarly, the κ-division is defined by: which reduces to the standard division, x/y, in the limit of κ → 0. By utilizing this κ-product, the κ-factorial, n! κ , with n ∈ N is defined by: Now, we come to the Stirling approximation of the κ-factorials.For sufficiently large n, the summation is well approximated with the integral as follows: Clearly, Equation (10) reduces to the standard Stirling approximation in the limit of κ → 0 as: Next, the κ-multinominal coefficient is defined by utilizing the κ-product and κ-division as follows: where we assume: In the limit of κ → 0, Equation ( 12) reduces to the standard multinominal coefficient of Equation ( 1).Now, let us try to relate the κ-multinominal coefficients with the κ-entropy.Taking the κ-logarithm of Equation ( 12) and applying the κ-Stirling approximation leads to: we then obtain: From this relation, we see that the above naive approach fails.Since the right hand side of Equation ( 15) consists of the two terms with different factors (one is proportional to n 1+κ /(1 + κ) and the other is proportional to n 1−κ /(1 − κ)), this cannot be proportional to the κ-entropy, which can be written by:

Introducing a New κ-Product
In order to overcome the above difficulty, we introduce another kind of κ-product based on the following function defined by: We here call it the κ-generalized unit function, since in the limit of κ → 0, it reduces to the constant function u {κ} (x) = 1.The basic properties of u {κ} (x) are as follows.
When κ = 0, the inverse function of u {κ} (x) can be defined by: In [14], the canonical partition function associated with the κ-entropy is obtained in terms of this u {κ} function.Note that: the two kinds of the κ-deformed functions (ln {κ} (x) and exp {κ} (x); u {κ} (x) and u −1 {κ} (x)) are thus associated with each other.This can be seen from the following relations: Now, by utilizing these functions, a new κ-product is defined by: Similarly, the corresponding κ-division is defined by: for any pair of the real numbers, x and y, such that x κ +x −κ 2 − y κ +y −κ 2 ≥ 1, and κ = 0.
It is worth noting that the new κ-product operator, ⊙ κ (the κ-division operator, ⊖ ÷ κ ), has no corresponding product (division) operator in the standard case of κ = 0.In other words, neither the product operator, ⊙ κ , nor the division operator, ⊖ ÷ κ , is a deformed (or generalized) one, and it has meaning only when κ = 0.In addition, unless κ = 0, this new product satisfies: There exists no real unit element on ⊙ κ .Therefore, real numbers and this product constitute a semi-group.Note also that x ⊖ ÷ κ x = 1, because u −1 {κ} (0) does not exist by the definition of Equation (20).However, we see that the following identity holds.
x ⊙ κ y ⊖ ÷ κ x = y, (x, y > 1, and κ = 0) Using the new κ-product, the associated κ-factorial can be introduced as: Similar to Equation (10), the κ-Stirling approximation can be obtained as: Furthermore, the corresponding κ-multinominal coefficient can be defined by: Applying the above Stirling approximation to Equation (31), we obtain: This is the complemental relation to Equation (15), and by combining Equations ( 15) and (32), we have: We thus obtain the final result: Note that since: as shown in the following, Equation (34) reduces to the standard case of Equation ( 2) in the limit of κ → 0.
and the κ-exponential function can be expanded in the form: Of course, in the limit of κ = 0, the κ-generalized gamma function, Γ κ (x), reduces to the standard gamma function, Γ(x); consequently, the κ-generalized factorial, n κ !, reduces to the standard factorial, n!.Since: by using integration by parts, the Γ κ (x) can be also expressed as: and by using the standard Γ(x), as: By integrating by parts, we obtain the property: which is equivalent with: The κ-generalized factorial also can be expressed as: Recently, Díaz and Pariguan [16] introduced the k-generalized gamma function and the Pochhammer k-symbol (x) n,k , which is defined by: For Re(x) > 0, the k-generalized gamma function is given by: (x) n,k ≡ x(x + k)(x + 2k) • • • (x + (n − 1)k)(49)for x ∈ C, k ∈ R and n ∈ N. By setting k = 1, one obtains the standard Pochhammer symbol, (x) n , which is also known as the raising factorial:(x) n = x(x + 1)(x + 2) • • • (x + n − 1)