{'fraction_non_alphanumeric': 0.046771349399613735, 'fraction_numerical': 0.016234220617460184, 'mean_word_length': 4.608791208791208, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Statistical field theories deformed within different calculi

Within framework of basic-deformed and finite-difference calculi, as well as deformation procedures proposed by Tsallis, Abe, and Kaniadakis to be generalized by Naudts, we develop field-theoretical schemes of statistically distributed fields. We construct a set of generating functionals and find their connection with corresponding correlators for basic-deformed, finite-difference, and Kaniadakis calculi. Moreover, we introduce pair of additive functionals, whose expansions into deformed series yield both Green functions and their irreducible proper vertices. We find as well formal equations, governing by the generating functionals of systems which possess a symmetry with respect to a field variation and are subjected to an arbitrary constrain. Finally, we generalize field-theoretical schemes inherent in concrete calculi in the Naudts spirit.


Introduction
In the course of the complex system investigations, vast variety of statistical theories has been developed [1,2,3,4,5,6,7,8,9,10]. The approaches related are based on the principle peculiarity of the statistical behavior of complex systems which is known to be their complicated dynamics being spanned in the fractal phase space governed by long-range interaction or long-time memory effects [11,12,13,14]. These complications have been overcome within framework of the standard statistical approach [15] by means of modification of the Boltzmann-Gibbs distribution due to deformations of the exponential function. Formally, the approaches proposed are based on using the extremum principle for a deformed entropic forms where the ordinary logarithm function is substituted with some versions modified according to Tsallis [1], Abe [4], Kaniadakis [5], Naudts [6], basic deformation procedure [10], et cetera. To our knowledge, the only effort along a direct using the field-theoretical method for development of a statistical scheme has been attempted in the work [16] for the Tsallis thermostatistics. The present article is undertaken with the purpose to generalize the standard statistical field theory [17,18] within different versions of calculi elaborated to this moment.
Historically, the first example of such a calculus gives the basic-deformed calculus (q-calculus, in other words) which has been originally introduced by Heine and Jackson [19,20] in the study of the basic hypergeometric series [21,22]. It appears the q-calculus does not only play a central role in the the quantum groups and algebra belonging to mathematical branches, but have a deep physical meaning [23,24]. In this context, the studying q-deformed bosons and fermions shows that thermodynamics can be built on the formalism of q-calculus where the ordinary derivative is substituted by the use of an appropriate Jackson derivative and q-integral [25]. Moreover, being based on a scale transformation related to the Jackson q-derivative and q-integral, the basic-deformed calculus is very well suited to describe multifractal sets [26,27]. Displaying critical phenomena of the type of growth processes, rupture, earthquake, financial crashes, these systems reveal a discrete scale invariance with the existence of log-periodic oscillations deriving from a partial breakdown of the continuous scale invariance symmetry into a discrete one -as occurs, for example, in hierarchical lattices [28,29,30,31].
Concerning the formalism on which is based our method, one needs to point out that it reduces to using a generating functional which presents the Fourier-Laplace transform of the partition function from the dependence on the fluctuating distribution of an order parameter to an auxiliary field [17,18]. Due to the exponential character of this transform determination of correlators of the order parameter is provided by differentiation of the generating functional over auxiliary field. As was mentioned above, fractality of the phase space of complex systems causes a deformation of both exponential function and integral itself in the Fourier-Laplace transform, so that the simple procedure of the differentiation of the generating functional becomes inconsistent. Thus, main problem of our consideration reduces to finding derivation operators, whose action keeps form of deformed exponentials giving eigen numbers related (within the basic-deformed calculus, such derivative reduces to the Jackson one, while eigen number is presented by basic-number [32]). This paper is devoted to development of the fieldtheoretical scheme basing on possible generalizations of both Fourier-Laplace transform and derivative operator related. The work is organized as follows. In Section 2, we yield a necessary information from the theory of both basic-deformed calculus and finite-difference calculus (hcalculus); moreover, we consider as well main peculiarities of the deformation procedures according to Tsallis, Abe, Kaniadakis, and Naudts. Sections 3-5 are devoted to construction of the generating functionals and finding their connection with related correlators for basicdeformed, finite-difference, and Kaniadakis calculi. Within the simplest harmonic approach, we find the partition functions and the order parameter moments as functions of the deformation parameters. Moreover, we introduce pair of additive functionals whose expansions into deformed series yield both Green functions and their proper vertices; as well, we find formal equations governing by the generating functionals of systems, which possess a symmetry with respect to a field variation and are subjected to an arbitrary constrain. In Section 6, we generalize the field-theoretical schemes elaborated in previous Sections. Section 7 concludes our consideration and Appendix A contains calculations related to deformed Gamma functions.

Preliminaries
We start by referring to the standard definition of the dual pair of the basic exponentials in form of the Taylor series [32] These series are seen to be determined by factorials [n] q ! = 1 · 2 · . . . · [n − 1] q · [n] q of the basic numbers [n] q := q n − 1 q − 1 (2) with the limit [n] q→1 = n due to which the functions (1) transform into the ordinary exponential at q → 1.
The mutual complementarity of the basic exponentials (1) is apparent from the multiplication rule [32] E q (x)e q (y) = e q (x + y) according to which The principle peculiarity of the exponentials (1) is to keep their forms under action of the Jackson derivative Really, one has for arbitrary constants a and b [32] D q In this way, for arbitrary functions f (x) and g(x) the Leibnitz rule reads: Let us write as well the useful equations e q (q n x) = 1 + (q − 1) x n q e q (x), following from the definition (6) and the equalities (7) at a = 1 and b = 0. Here, we use the basic-deformed binomial [32] (x + y) n q := (x + y)(x + qy) . . . (x + q n−1 y).
In addition to the basic exponentials (1) being invariant with respect to action of the Jackson derivative (6), its eigen-functions are known to represent the homogeneous functions determined with the property where an exponent q plays the role of the self-similarity degree, λ is an arbitrary factor playing the role of deformation of self-similar systems for which the homogeneous functions are the basis of the statistical theory related [31]. To this end, the eigen-values of the Jackson derivative (6) determined on the set of the homogeneous functions represent the basic numbers (2): Apart from invariant action of the Jackson derivatine D q x , the exponentials (1) are persistent as well under action of the basic integral I q x , whose property D q x I q x = 1 relates to the explicit definition [32] I q Meaning D q x and I q x as transformation operators of the Lee group, let us introduce the generators related: Here, the pair of the basic logarithmic functions ln q (x) and Ln q (x) is used as inverse functions to the exponentials (1). By using the properties u q x + j q x = 0 and U q x + J q x = 0, one obtains the functional expressions of the basic integral through the Jackson derivative: Let us list now main cases of deformations that complete the basic deformation related to the exponentials (1). Above, we have demonstrated the dual pair of these exponentials is determined by the numbers [n] q and [n] 1/q defined by Eq. (2). In the case of the symmetrized q-calculus, the corresponding number is obeyed the condition [n] 1/q = [n] q , so that the dual exponential E q (x) = e 1/q (x) coincides with the original one, e q (x). The symmetrized q-calculus is a basis of the Abe statistics [4]. The third example gives the h-exponential that is specified by the expression [32] e h (x) : This function is inverse to the h-logarithm with ln(x) being the ordinary logarithm function. It is easy to convince the expression (17) may be presented in form of the series (1) if the basic numbers (2) are substituted by the h-numbers [n] h := hn ln (1 + h) .
The dual h-exponential can be defined as E h (x) := e −h (x) to obey the trivial rule instead of Eq. (4). Setting the symmetry with respect to change of the h sign, we obtain the self-dual exponential , with whose consideration we restrict ourselves further. In the limit h → 0, the h-calculus reduces naturally to the usual one.
The h-exponential (17) is obviously invariant with respect to action of the h-derivative [32] D h However, the properties (7) take the form complicated with the factor d h (a). Although this factor has the ordinary limit d h→0 (a) → a, action of the hderivative (21) on the exponential (17) does not obey a condition of the type (7) for arbitrary values of a constant a. To restore such a condition we shall use the definition of the h-derivative instead of Eq. (21). Being applied to the exponential (17), this derivative ensures obviously the first propertiy (7) with index q substituted by h. Respectively, the h-integral, being inverse to the derivative (23), is defined as Quite different example represents the Tsallis exponential exp q (x) := [1 + (1 − q)x] 1 1−q (25) characterized by the deformed number Here, the dual number [n] 1/q relates to the exponential whose inverse value is proportional to the escort probability being the basis of the Tsallis thermostatistics [3,14]. It is principally important for our aims the following: i) the Tsallis exponential (25) is not invariant with respect to the Jackson derivative (6), whereas action of the ordinary derivative gives d dx exp q (x) = exp q q (x); ii) the dual exponential (27) is obeyed the condition Exp 1/q q (−qx) exp q (x) = 1 that does not coincide with the condition (5). In the limit q → 1, the Tsallis calculus reduces to the ordinary one, while it coincides with the h-calculus at arbitrary values of the deformation parameter q = 1−h/x. It is worthwhile to stress, however, the Tsallis exponential increases with the x-growth according to the power law (25), while the hexponential (17) varies exponentially. The statistical field theory based on the Tsallis calculus has been developed in work [16].
As the following example, we consider the case of the Kaniadakis deformation when exponential and logarithm functions are defined as follows [5]: Here, the deformation parameter κ belongs to the interval (−1, 1) and the limit κ → 0 relates to the ordinary functions exp(x) and ln(x). The exponential (28) is self-dual function in the the sense that it is obeyed the condition exp κ (x) exp κ (−x) = 1 of the type (5). However, the multiplication rule (4) takes the form [33] exp where the sum is deformed as Remarkably, the rule of the type (30) takes place also for the Tsallis q-calculus where the deformed sum (31) is written as [37] x ⊕ q y := x + y + (1 − q)xy.
Unfortunately, the exponential (28) may not be presented in the form of any deformed series (1) with a number of the type (2). However, it is easy to convince that this exponential is invariant with respect to action of the derivation operator [33] Moreover, for arbitrary constant a one obtains instead of Eqs. (7) and (22). The integration operator being inverse to the derivative (33) is defined in the relativistic form [33] I The derivative property (34) accompanied with the integral definition (35) will be shown to be formal statement for development of the field-theoretical scheme based on the Kaniadakis calculus.
Above considered examples show that a generalized calculus can be built as a result of the following steps: 1. Choose a deformed exponential e λ (x) and find its dual form E λ (x) to be obeyed the multiplication rule If above exponentials may be expanded into the Taylor series of the type (1), their choice is fixed by numbers [n] λ generalizing the expressions (2), (16), and (19) with parameter q being substituted with a deformation λ. In the case of the type of both Tsallis and Kaniadakis deformations, more convenient to use a self-dual exponential obeying the multiplication rule defined by specifying deformed sum x λ ⊕ y [see, for example, Eqs. (31) and (32)].

Define a deformed differentiation operator D λ
x type of the Jackson derivative (6) according to the condition that this operator keeps invariant forms of the generalized exponentials e λ (x) and E λ (x). 3. Introduce a deformed integration operator according to the definitions generalizing Eqs. (15) (here, deformed logarithmic functions ln λ (x) and Ln λ (x) are defined to be inverse to the generalized exponentials related).
As a result, we achieve the position to develop a statistical field theory that is based on the use of a generating functional being a generalization of the characteristic function [17,18]. This function is known to be presented by the Fourier-Laplace transform of the probability distribution p(x). The key point is that deformed exponential standing within integrand of the characteristic function (43) is eigen-function of the deformed derivative operator D λ j with eigen value d λ (x). As a result, multiple differentiation of this function over auxiliary variable j keeps its exponential form to yield the moments of an order parameter d λ (x).

Basic-deformed statistics
Let us consider a statistical system, whose distribution over states x = {r a , p a } in the phase space of particles a = 1, . . . , N , N → ∞ with coordinates r a and momenta p a is determined by a Hamiltonian H = H (x). We are interested in study of the coarse space distribution φ(r) of an order parameter φ. Within the coarse grain approximation, thermostatistics of the deformed system is governed by the partition functional Here, d q x = (q − 1)x stands for the basic-deformed differential, S = S{φ} is an effective action, and one takes into account also that thermostatistical distribution of a basic-deformed system is proportional to the q-deformed exponential e q [−βH (x)] with the inverse temperature β measured in the energy units [10].
The principle peculiarity of the definition (45) is that both thermostatistical exponential and integral over the phase space are the basic deformed ones. To this end, we need to use the basic-deformed Laplace transform where the last equation is written with accounting the property (4). For the sake of simplicity, we use the lattice representation to describe the coordinate dependence by means of According to the rules (7) the n-fold differentiation of the last expression for the generating functional (46) yields The right hand site of this equality determines the correlator where the coefficient is inversely proportional to the partition function Combination of the last equalities allows for one to express an arbitrary correlator through the basic-deformed derivatives of the generating functional (46): Within framework of the harmonic approach, effective action takes the deformed parabolic form where the deformed binom (10) is used and ∆ 2 stands for the inverse curvature. To apply the rule (4) for the basicdeformed exponential in the generating functional (46) let us suppose the symmetry with respect to substituting the deformation parameter q by the inverse value 1/q. Then, it is convenient to separate whole lattice into odd sites i and even ones i and use Eq. (4) for each of couples i , i . To this end, the exponential in the partition function (49) is transformed as follows: where square brackets denote the integer of the fraction N/2. 1 As a result, the generating functional (46) takes the multiplicative form As show simple calculations in Appendix A, each of multipliers related to one site is determined by the expression where the basic-deformed γ-function is defined by the first equation (A.1). Respectively, specific partition function z q (J = 0) reads: As shows Figure 1a, the dependence of this function on the deformation parameter has in logarithmic axes symmetrical form with respect to the maximum point q = 1.
According to the definition (50), within the harmonic approach the order parameter is determined as where the second relation (9) is used. In similar manner, cumbersome but simple calculations give the field variance Thus, the basic-deformed distribution of free fields has zero moment of the first order and the variance, being proportional to the inverse curvature of the related action (51) and dependent on the deformation parameter linearly.
To develop a deformed perturbation theory one ought, as usually, to pick out an unharmonic contribution V = V {φ} in total action S = S (0) + V [18]. Then, the basicdeformed exponential can be written as follows: As a result, the formal expansion of the exponential E q (−V ) in power series with consequent use of the differentiation 1 Generally speaking, a statistical ensemble might comprise of odd number of particles N that must arrive at unmatched factor in products (52). However, this factor is negligible within the thermodynamic limit N → ∞. rules (7) allows for one to present the generating functional (46) in the convenient form Further, making use of the perturbation scheme with implementation of related diagram technics is straightforward [18]. Moreover, the thermodynamic limit N → ∞ allows for one to use the Wick theorem to express higher correlators through the variance (56).
Similarly to the ordinary field scheme [18], an inconvenience of the above approach is that the generating functional (46) is non-additive value. To escape this drawback one should introduce the Green functional being deformed logarithm of the functional (46). It worthwhile to note the function of the deformed logarithm has not an explicit form to be defined by the inverse exponential function Z q = e q (G q ) given by the first series (1). Since the Green functional (59) depends on an auxiliary field J, it can be more convenient to use a conjugate functional Γ q = Γ q {φ}, whose dependence of the initial field φ is provided by the Legendre transformation The pair of the functionals G q {J} and Γ q {φ} plays the role of conjugated potentials, whose basic-deformed variation yields the state equations Being analytical functional, these potentials can be presented by the following series: To this end, related kernels G i1...in reduce to n-particle Green function and its irreducible part, respectively. Within the diagram representation, these kernels are depicted in Figure 2.
Following the standard field scheme [18], we show further that the generating functional (46) obeys some formal relations. The first displays a system symmetry with respect to the basic-deformed variation in the form given by an analytical functional f i {φ} in the limit q → 0. Due to this variation the integrand of the last expression of the functional (46) is transformed in the following manner: where the sum over repeated indexes is implied and the second expansion (1) is taken into account. On the other hand, the Jacobian determinant appearing due to passage from φ to φ + δ q φ gives the factor 1 + (∂f i /∂φ i ) q . As a result, collecting multipliers which include the infinitesimal value q , one obtains from the invariance property of the generating functional (46): (66) Here, we use the first of the state equations (61) for operator representations of the type . At condition f i {φ} = const, the equation (66) takes the simplified form, following directly from the generating functional (46) after variation over the field φ.
The second of above pointed equations allows for one to take into account an arbitrary condition F j {φ} = 0, j = 1, 2, . . . for the set of fields {φ i }, i = 1, 2, . . . , N to be found. Taking into account of this condition is achieved by inserting the δ-functional δ{F } into the integrand of the expression (46) that results in introducing the prolonged form Then, variation over an auxiliary variables λ j , j = 1, 2, . . . yields the desired result In comparison with the standard field theory [18], the principle peculiarity of the equalities (66) and (68) is that they contain the Jackson derivative D q J instead of the ordinary variation δ/δJ.

Finite-difference statistics
Within framework of the finite-difference statistics, the field scheme is based on the definitions (17)- (20), (23) and (24) inherent in the h-calculus [32] to be developed in analogy with consideration stated in the previous section. With using the h-exponential (17), the partition functional takes the form of the h-integral instead of the expression (45). Respectively, the generating functional (46) is written as Similarly to action of the Jackson derivative on the basic exponential, the differentiation operator (23) gives the correlator (48) with q substituted by h in the following form [cf. Eq. (50)] As in Eq. (51), the harmonic action is determined with the inverse curvature ∆ 2 . Then, the generating functional (70) is expressed by the product of the type (53) with the one-particle factor (A.8). Taking into account the property (A.7) of the h-gamma function, one obtains the expression reducing to the standard form in the limit h → 0. Moreover, the specific partition function related to J = 0 takes the non-deformed value z 2π∆ for arbitrary parameters h. According to the definition (71), the mean value of h-deformed free fields equals Respectively, the variance related is written as instead of Eq. (56). Thus, similarly to the basic-deformed distribution the h-deformed free fields have zero moment of the first order and the variance being proportional to the inverse curvature of the action (72) related. However, dependence of the variance (77) on the deformation parameter appears to be more strong than the linear dependence (56) related to the case of the basic deformation (see Figure 3a).
The h-deformed perturbation theory is built in the complete accordance with the scheme stated in the previous section, with the only difference that the dual qexponential E q (x) should be substituted with the self-dual h-exponential e h (x). Along this line, making use of the symbolic perturbation expansion of the type (57) yields the generating functional instead of Eq. (58). As well, application of the diagram technics and the Wick theorem appears to be straightforward. Moreover, following to the standard line [18], one should supplement the non-additive functional (70) by the Green functional G h := ln h (Z h ) where the h-logarithm is defined as (18) to be inverse to the h-exponential (17).

Kaniadakis statistics
Taking into account close likeness between basic-deformed, Tsallis and Kaniadakis statistics, we present the latter basing on the field-theoretical schemes developed in Section 3 and Ref. [16]. In so doing, one needs to substitute the Tsallis deformed exponentials (25) by the Kaniadakis ones (28) and take into account the multiplication rule (30) defined with the deformed sum (31). As a result, the generating functional (46) takes the form According to the property (34), the correlator [cf. with Eq. (50)] is determined by the partition function Z κ = Z κ {J = 0} and the Kaniadakis derivative (33). Then, with using the harmonic action the generating functional (79) reduces to the product of the one-site factors The specific partition function z (84) According to Eqs. (80) and (83), the first moment is φ (0) κ = 0, while the definition of the free field variance φ 2 (0) κ is achieved by inserting the φ 2 factor into the integrand of the integral (84) and dividing by z (0) κ . As shows Figure 1b, dependence of the one-site partition function (84) on the deformation parameter κ has a symmetrical form with respect to the point κ = 0. Characteristically, the |κ| arising results in the z (0) κ increase at small values ∆ 2 of the inverse curvature of action (81), while the ∆ growth transforms the concave curve of the dependence z (0) κ (κ) into the convex one. What about the dependence φ 2 (0) κ (κ) for the variance of the κ-deformed free fields, Figure 3b visualizes more complicated curve: first, the |κ| growing results in the φ 2 (0) κ decrease from the value φ 2 (0) κ=0 = ∆ 2 , after that the field variance increases before an anomalous value φ 2 (0) |κ|=1 > ∆ 2 .

Generally deformed statistics
The examples considered in Sections 3-5 and Ref. [16] show the generalization of the deformed statistics should be performed along the two different lines. The first generalizes the basic-deformed and h-statistics, the second makes the same for statistics of the type proposed by Tsallis and Kaniadakis. Such a partition is stipulated by the principle difference between the multiplication rules (4) and (30). In the first case, this rule is provided by means of finding an exponential E λ (x) being dual to the initial one e λ (x) (this case is implemented for the basic-and hcalculi, and the latter relates to the self-dual exponential); the second class requires to deform the sum standing in the exponent of r.h.s. of Eq. (41) according to rules of the types (31) and (32). Let us consider above cases separately.
Following to the method developed in the end of Section 2, generalization of both basic-deformed and h-statistics is carried out in the straightforward manner. However, as shows comparison of the considerations stated in Sections 3 and 4, to ensure the passage to the expression that contains the ordinary sum of exponents one follows to use the deformed Laplace transform with the dual exponential E λ (x) ⇒ E q (x), being inherent in the basic-deformed calculus with q = λ, or the expression with the initial exponential e λ (x) ⇒ e h (x), taking place in the h-deformed calculus with h = λ [cf. Eqs. (46) and (70)]. In all above cases, the n-fold derivative yields Here, we take into account the differentiation rule where an eigen-value η λ (φ i ) is determined by action of a generalized derivative D λ Ji with respect to a auxiliary field J i (in the simple cases of both basic-and h-deformed calculi, one has η λ (φ i ) = φ i , while the Tsallis calculus relates to η λ (φ i ) = ln 2−λ (φ i ) [16]). As a result, the correlator with the partition function is expressed in the form Within the harmonic approach, the action (51) is written in the square-law form where a field set {φ i } is distributed with the unit variance and λ-deformed square (φ i ) 2 λ and number [2] λ are used instead of Eqs. (2) and (10). Then, the generating functional (86) takes the form of the type (53). Respectively, for the generating functional (87) one obtains Here, each of multipliers related to one site is determined by the expression where ∆ λ is a λ-deformed variance taking the value ∆ λ0 = 1 in the non-deformed limit λ → λ 0 ; in turn, λ-deformed π-number is defined by equation of the type (A.5) with λ standing instead of q (as pointed out above, a dual λexponential E λ (x) should be used in the case of the type qcalculus and a self-dual λ-exponential e λ (x) -for the class of h-type calculus). By this, the specific partition function z Similarly to the q-and h-statistical field theories, the mean value of free fields equals η(φ) (0) λ ∝ J| J=0 = 0, while the variance η 2 (φ) (0) λ appears to be monotonically increasing function of the deformation parameter λ. Calculation of explicit form of dependences (96) and (97), as well as the variance η 2 (φ) (0) λ of order parameter η = η(φ) related to the derivation rule (89) needs to specify a concrete form of the λ-deformed exponentials.
The perturbation theory is based on the equation type of (57) and (58), as well as the diagram technics and the Wick theorem are built similarly to above considered field-theoretical schemes. Moreover, the additive generating functional related to the non-additive ones, (86) and (87), is expressed by the Green functional G (0) λ {J} in the form (59). The passage from this functional, dependent on an auxiliary field J, to the conjugate functional Γ λ = Γ λ {φ}, being a functional of an order parameter φ, is achieved by the Legendre transformation type of (60), while the state equations have the form (61). Respectively, series of sort (62) and (63) have kernels that reduce to the n-particle Green function and its irreducible part. Within the diagram representation, these kernels look like depicted in Figure 2. What about the formal equations for the generating functionals (86) and (87), they take the forms (66), (67), and (68) to display a system symmetry with respect to the λ-deformed variation δ λ φ i (x) and take into account an arbitrary condition F j {φ} = 0, j = 1, 2, . . . for the set of fields {φ i }, i = 1, 2, . . . , N . As pointed out above, in all above expressions the index q should be substituted by the subscript λ, and the λexponentials e λ (x) and E λ (x) should be used instead of the q-exponentials e q (x) and E q (x).
Finally, we state main principles of generalization of the Kaniadakis-type statistics. As was stressed in the beginning of this Section, the corner stone of the Kaniadakis calculus is that the multiplication rule (30) is defined by deformed sum of the type (31). As a result, the generating functional (79) takes the form determined by a generally deformed sum standing in the last exponent. Respectively, the correlator (80) is written as follows: with the partition function Z λ = Z λ {J = 0}. Instead of Eq. (89), we take into account here the differentiation rule D λ(a) x e λ (ax) = η λ (a)e λ (ax) with an eigen-value η λ (a) and a deformation λ(a), being defined by action of a generalized derivative D λ(a) x with respect to a variable x at arbitrary constant a (for the Kaniadakis calculus, one has η λ (a) = a and λ(a) = λa). By analogy with above considered field-theoretical schemes, one finds partition function, perturbation theory, diagram technics, additive and conjugated Green functional, many-particle Green functions and their irreducible parts, as well as formal equations for the generating functional of systems that display a symmetry with respect to field variation and have some constraints.

Concluding remarks
Before a discussion of the results obtained we should stress uppermost that quantum algebra and quantum groups, on whose basis our approach is founded, have been the subject of intense research in different fields of the q-deformed quantum theory (see [38] and references therein). Moreover, using the q-deformed algebra has allowed to develop multifractal theory [39,27] and thermostatistics of deformed bosons and fermions [25]. In our consideration, we restrict ourselves with generalization of the only classical thermostatistics, whose version has been developed first in work [10]. The principle peculiarity of the basic-deformed distribution arising in this model is to exhibit a cut-off in the energy spectrum which is generally expected in complex systems, whose underlying dynamics is governed by long-range interactions.
It is worthwhile to note in this connection the deformed distribution proposed by Tsallis [1] that is characterized by the power-law asymptotic behavior. The Tsallis picture is known to be inherent in self-similar statistical systems, whose field theory has been built by using both Mellin transform of the Tsallis exponential and Jackson derivative [16]. Contrary to the examples considered in Sections 3-6, a fluctuating order parameter of self-similar systems has non-zero mean value that reduces to the Tsallis deformed logarithm of the amplitude of a hydrodynamic mode. Formally, this is caused by using the Mellin transform with the power-law kernel φ J = exp[J ln(φ)] instead of the Fourier-Laplace one with the exponential kernel exp(Jφ).
Above, we develop field-theoretical schemes founded on the basis of basic-deformed and finite-difference calculi [32], as well as within framework of deformation procedures proposed by Tsallis, Abe, Kaniadakis, and Naudts [1,4,5,6]. We construct generating functionals related and find their connection with corresponding correlators for basic-deformed, h-, and Kaniadakis calculi. Moreover, we introduce pair of additive functionals whose expansions into deformed series yield both Green functions and their irreducible proper vertices, as well as find formal equations, governing by the generating functionals of systems which possess a symmetry with respect to a field variation and are subjected to an arbitrary constrain. Finally, we generalize in the Naudts spirit the field-theoretical schemes inherent in concrete calculi.
Concerning the physical results above obtained, one should point out peculiarities of dependences of both onesite partition function and variance of free fields on de-formations (see Figures 1 and 3, respectively). In the case of basic deformation, the specific partition function has in logarithmic axes symmetrical form with respect to the maximum point q = 1 (by this, the ∆ 2 increase shifts up the curves of dependences related logarithmically equidistantly). For the h-deformation, the specific partition function takes non-deformed value. In the case of the Kaniadakis deformation, the dependence of the one-site partition function z (0) κ on the deformation parameter κ has a symmetrical form with respect to the point κ = 0 (by this, the κ growth results in the z (0) κ increase at small inverse curvature ∆ 2 of the effective action, while the ∆ growth transforms the concave curve of the dependence z (0) κ (κ) into the convex one). What about the correlators of free-distributed fields, the only moment of the second order takes non-zero values. For all distributions, this moment is proportional to the inverse curvature ∆ 2 of the action related to increase with the deformation parameter growth linearly in the case of the basic-deformed statistics and non-linearly rapidly for the h-statistics. More complicated behaviour takes place for the Kaniadakis deformation, when the variance related decreases first and increases then up to an anomalous value.
The authors are very grateful to the organizing committee of the NEXT-ΣΦ 2008 conference in Kolymbari, Crete, Greece. We thank especially G. Kaniadakis and A. Scarfone for sending us the special issue of the European Physical Journal containing several papers reported in this conference.

Appendix A
Within framework of the basic-deformed calculus, the pair of dual q-gamma functions is defined by the integrals where the upper limit equals 1 1−q for |q| < 1. These definitions are principle different from the introduced in Ref. [32] because we substitute the ordinary power function where deformed π-number π q has the limit π q→1 = π ≡ 3.14159 . . . The specific generating functional is calculated as: where the variable x = φ − ∆ 2 J is introduced.
In the case of the h-deformed calculus, the gamma function related appears to be proportional to the ordinary one Γ (α). By analogy with the calculations (A.6), the one-site generating functional is expressed as: