A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains

A q - C A P U T O F R A C T I O N A L G E N E R A L I Z A T I O N O F T S A L L I S E N T R O P Y : S E R I E S R E P R E S E N T A T I O N A N D N O N - N E G A T I V I T Y D O M A I N S A P R E P R I N T Matias P . Gonzalez Departamento de Física Univ ersidad Católica del Norte Antofagasta, Chile matias.gonzalez03@alumnos.ucn.cl Micolta-Riascos Bayron Departamento de Física Univ ersidad de Antofagasta Antofagasta, Chile bayron.micolta.riascos@ua.cl March 25, 2026 A B S T R AC T W e introduce a fractional generalization of Tsallis entropy by acting with a q -Caputo operator on the generating family P i p x i ev aluated at x = 1 . Concretely , we define S α q through the q -Caputo differinte gral of order 0 < α < 1 and deri ve a closed series representation in terms of the q -Gamma function. The construction is anchored at the ev aluation point, which ensures well-behaved limits: as α → 1 we recov er the standard Tsallis entropy S q . Finally we perform a numerical calculation to sho w the regions where the obtained q -fractional entropy S α q can be non-neg ativ e (or negati ve) through the fractional parameter α and the non e xtensiv e index q . Keyw ords Tsallis entropy · fractional calculus · q -calculus · Caputo deri vati ve · q -Gamma function 1 Introduction The Tsallis entropy , denoted as S q , has emerged as a successful tool to e xplain a wide range of phenomena, such as long-range interactions, correlations, and the fractal properties of phase space [ 1 , 2 ]. In this w ork, we aim to generalize Tsallis entropy using the framew ork of fractional calculus. The approach begins by considering the Jackson deriv ativ e D q , which is generalized through the use of the fractional Caputo deri vati ve C D α , resulting in what we define later in this work as a q -fractional Caputo deriv ative C D α q which is our main tool for the dev elopment of this work. First, we re view the Tsallis entrop y S q , its definition and the proceeding on ho w it can be deri ved from the Jackson deri v ative when applied to probabilities p x i (analogously as Tsallis) [ 2 ]. W e then introduce the Caputo fractional deriv ative and demonstrate its combination with the Jackson deri v ative, yielding the principal analytical tool for our study . W e deri ve a fractional generalization of Tsallis entropy by using techniques from q -fractional calculus. Finally , we show the non-neg ativity domain of the fractional Tsallis entropy for two equiprobable microstates. 2 Tsallis Entropy Tsallis statistics relies on the generalization of the Boltzmann-Gibbs entropy S BG into the Tsallis entropy S q [ 1 ]. This functional is the starting point for the non-extensi ve formalism and is defined by S q ≡ k 1 − P i p q i q − 1 , (1) where { p i } denotes the probabilities of the system’ s microstates, k is a constant analogous to Boltzmann’ s, and q is the nonextensi ve parameter which is a real number . For the limit case q → 1 , this expression reduces to the Boltzmann-Gibbs entropy , thereby recov ering the standard statistical-mechanical framew ork. Tsallis entrop y S q maintains structural properties of the Boltzmann-Gibbs entrop y S BG : for normalized distrib utions and q > 0 it is non-ne gativ e, and it attains its maximum for the uniform (equiprobable) distribution. The k ey dif ference lies in the composition A q -Caputo F ractional Generalization of Tsallis Entr opy: Series Repr esentation and Non-Ne gativity Domains rule. Whereas S BG is strictly additiv e, S q is non-additiv e (pseudo-additiv e). F or two statistically independent subsystems A and B with factorized joint probabilities, the total entropy satisfies S q ( A + B ) = S q ( A ) + S q ( B ) + 1 − q k S q ( A ) S q ( B ) , (2) The final term measures the departure from additi vity and is go verned by the nonextensivity index q ; it vanishes as q → 1 , recov ering the additiv e Boltzmann-Gibbs limit. This correction does not imply physical correlations between A and B , it reflects the ef fectiv e coupling introduced by the nonextensi ve entropy , a feature useful for modeling systems with long-range interactions, memory effects, or constrained phase space. A compact route to (1) uses the Jackson q -deriv ati ve, a finite-difference deformation of the ordinary deri vati ve [2]: D q f ( x ) ≡ f ( q x ) − f ( x ) q x − x . (3) Identifying the Jackson parameter with the entropic index q and applying D q to f i ( x ) = p x i at x = 1 , D q p x i    x =1 = p q i − p i q − 1 , (4) − k X i D q p x i    x =1 = − k P i p q i − P i p i q − 1 = k 1 − P i p q i q − 1 = S q , (5) where we used the normalization P i p i = 1 . In the limit q → 1 implies D 1 p x i   x =1 = p i ln p i , yielding the Boltzmann-Gibbs entropy S B G = − k P i p i ln p i . The last deriv ation will be the main road to the fractional generalization of Tsallis entropy . 3 q-Caputo derivati ve Fractional calculus is an extension of the concept of entire order calculus, in which we get a general operator called dif ferintegral operator of order α , D α . That new operator is a fractional deriv ative when α > 0 and is a fractional integral when α < 0 , defining the fractional deriv ative of ne gativ e orders as a fractional integral D − α ≡ I α , where I α is defined as I α f ( t ) = 1 Γ( α ) Z t 0 f ( τ )( t − τ ) α − 1 dτ . (6) Thus, fractional Caputo deriv ative [3] is defined as D α = D α − n D n = I n − α D n , where n ∈ Z and α ∈ R : C D α f ( t ) = 1 Γ( n − α ) Z t 0  d n f ( τ ) dτ n  ( t − τ ) n − α − 1 dτ , n = [ α ] + 1 . (7) This notion is also extended including q -calculus, and defines the q -integral of order α ∈ R and the q -Caputo deriv ative of order alpha as follows: I α q f ( z ) = 1 Γ q ( α ) Z t 0 ( t − q τ ) α − 1 q f ( τ ) d q τ , (8) C D α q f ( t ) = 1 Γ q ( n − α ) Z t 0  D n q f ( τ )  ( t − q τ ) n − α − 1 d q τ , (9) where Γ q ( z ) is the q -Gamma function [4] defined by Γ q ( z ) = Z 1 1 − q 0 x z − 1 E − q x q d q x, (10) E z q is also defined by E z q = ∞ X n =0 q n ( n − 1) / 2 z n [ n ] q ! , [ z ] q = 1 − q z 1 − q = 1 + q + · · · + q z − 1 , (11) where the first order q -deriv ati ve is giv en by Eq. (3) with d q f ( z ) = f ( q z ) − f ( z ) and d q z = ( q − 1) z as q -differentials. 2 A q -Caputo F ractional Generalization of Tsallis Entr opy: Series Repr esentation and Non-Ne gativity Domains 4 Derivation of the Fractional Tsallis Entr opy W e start re writing the Tsallis entropy: S q = − D q W X i =1 p x i      x =1 = − W X i =1  p q x i − p x i q x − x       x =1 = − W X i =1  e q x ln p i − e x ln p i q x − x       x =1 = − W X i =1 P ∞ m =0 q m x m (ln p i ) m m ! − P ∞ m =0 x m (ln p i ) m m ! q x − x !      x =1 = − W X i =1 ∞ X m =0 x m (ln p i ) m ( q m − 1) m ! x ( q − 1)      x =1 . (12) The m number is defined as a con ver gent geometric sum [5]: [ m ] q = 1 − q m 1 − q = 1 + q + . . . + q m − 1 , (13) thus, the last expression becomes S q = − W X i =1 ∞ X m =0 [ m ] q m ! (ln p i ) m x m − 1      x =1 . (14) In addition, using the definition of the q -Caputo deriv ati ve, we propose the next generalization: S α q = − C D α q W X i =1 p x i      x =1 , 0 < α < 1 . (15) This carries to S α q = − I 1 − α q D q W X i =1 p x i      x =1 = − I 1 − α q W X i =1 ∞ X m =0 [ m ] q m ! (ln p i ) m x m − 1      x =1 = − W X i =1 ∞ X m =0 [ m ] q m ! (ln p i ) m Γ q ( m ) Γ q ( m − α + 1) x m − α      x =1 , (16) using [ m ] q = Γ q ( m + 1) Γ q ( m ) we obtain: S α q = − W X i =1 ∞ X m =0 Γ q ( m + 1) m !Γ q ( m + 1 − α ) x m − α (ln p i ) m      x =1 , (17) then taking x = 1 we arriv e to the form of the q -caputo Tsallis entropy S α q = − W X i =1 ∞ X m =0 Γ q ( m + 1) m !Γ q ( m + 1 − α ) (ln p i ) m . (18) For more details about q -calculus and fractional calculus see [6]. 3 A q -Caputo F ractional Generalization of Tsallis Entr opy: Series Repr esentation and Non-Ne gativity Domains 5 Limit case α → 1 Now we w ant to prove analytically that if we set α → 1 S q is recov ered. Starting from (18) we have S α → 1 q = − W X i =1 ∞ X m =0 Γ q ( m + 1) m !Γ q ( m ) (ln p i ) m , (19) where the identity ∞ X m =1 [ m ] q m ! t m = e q t − e t q − 1 , (20) is used by defining t = ln p i , the m = 0 term in the sum is equal to zero and using [ m ] q = Γ q ( m + 1) / Γ q ( m ) we get S α → 1 q = − W X i =1 e q ln p i − e ln p i q − 1 = − W X i =1 p q i − p i q − 1 , (21) using P W i =1 p i = 1 it simplifies to S α → 1 q = − P W i =1 p q i − 1 q − 1 = 1 − P W i =1 p q i q − 1 , (22) which recov ers Tsallis entropy (1) assuming natural units k = 1 . 6 Negativity Figure 1: Non-negati vity domain of the fractional Tsallis entropy S α q for two equiprobable microstates ( W = 2 , p i = 1 2 ). Shaded (blue) region corresponds to S α q > 0 , grey region to S α q < 0 , and the black curve marks the implicit boundary S α q = 0 . The boundary is obtained by e valuating the exact series definition in Eq. (18) with the q -Gamma coef fi cients, sweeping the parameters space ( α, q ) with 0 < α < 1 and 0 ≤ q ≤ 2 . Tsallis entropy is non-neg ativ e, S q ≥ 0 [ 2 ], whereas our fractional construction S α q can reach negati ve v alues. W e ev aluate (17) numerically for two equiprobable states and map the ( α, q ) domain 0 < α < 1 , 0 ≤ q ≤ 2 , (23) 4 A q -Caputo F ractional Generalization of Tsallis Entr opy: Series Repr esentation and Non-Ne gativity Domains identifying the regions where non-negati vity holds (or is violated), see Fig. 1. The lack of a positivity guarantee follows from the structure of (17): for 0 < p i < 1 one has (ln p i ) m = ( − 1) m | ln p i | m , and with Γ q ( m + 1) m ! Γ q ( m + 1 − α ) > 0 , the series reorganizes as a dif ference of two positiv e sums, S α q = W X i =1       X m odd Γ q ( m + 1) m ! Γ q ( m + 1 − α ) | ln p i | m | {z } > 0 − X m even Γ q ( m + 1) m ! Γ q ( m + 1 − α ) | ln p i | m | {z } > 0       , (24) which can render S α q < 0 for certain ( α, q ) because the “odd minus ev en” combination is not necessarily positiv e. In particular , the m = 0 term contributes − 1 / Γ q (1 − α ) < 0 , explaining the onset of ne gativity near the probability-domain boundaries and for small α . 7 Conclusions In this work, we generalized the usual Tsallis entropy S q by means of a fractional definition of Jackson’ s deriv ativ e, namely the q -Caputo deriv ativ e C D α q with 0 < α < 1 , thereby obtaining a fractional form S α q . W e studied the limit case α → 1 , which recov ers the standard Tsallis entropy . Concerning the properties, non-negati vity is not satisfied for ev ery combination ( α, q ) in the parameter space, which leads to forbidden re gions if non-negativity (positi vity) is required to be preserv ed. On the other hand, pseudo-additivity is a property that needs to be formally pro ved in future work As a future outlook, we propose this entropy as a basis for upcoming applications and also with the purpose of further strengthening its formalism o ver time, showing that q -calculus and fractional calculus can be combined in a consistent way within the same framework. In particular , the fractional Tsallis entropy S α q may serve as a starting point to explore new models in non-extensi ve statistical mechanics, information theory , and complex systems, where memory effects and non-locality play a rele vant role. W e expect that, as this formalism is refined and confronted with concrete physical and mathematical applications, it will help to clarify the interplay between non-extensi vity , q -calculus, and fractional operators, and to moti vate further generalizations in related fields. References [1] Constantino Tsallis. Possible Generalization of Boltzmann-Gibbs Statistics. J . Statist. Phys. , 52:479–487, 1988. [2] Constantino Tsallis and Ugur T irnakli. Nonadditi ve entropy and nonextensi ve statistical mechanics – some central concepts and recent applications. Journal of Physics: Confer ence Series , 201:012001, February 2010. [3] Alexander I. Zhmakin. A compact introduction to fractional calculus, 2022. [4] Alberto De Sole and V ictor Kac. On inte gral representations of q-gamma and q-beta functions, 2003. [5] Mahmoud H. Annaby and Zeinab S. Mansour . q-F ractional Calculus and Equations . Springer Berlin, Heidelberg, 2012. [6] Y ing Sheng and Tie Zhang. 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