An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function

A significant advance in characterizing the nature of the zeros and organizing theMittag-Leffler functions into phases according to their behavior is presented. Regions have been identified in the domain of α and β where the Mittag-Leffler functions E α,β (z) have not only the same type of zeros but also exhibit similar functional behavior, and this permits the establishment of an α-β phase diagram.


Introduction
The Mittag-Leffler (ML) function  , () defined by is a generalization of the exponential function and plays a fundamental role in the theory of fractional differential equations with numerous applications in physics.Consequently, books devoted to the subject of fractional differential equations (i.e., Podlubny [1], Magin [2], Kilbas et al. [3], and Mainardi [4]) all contain sections on the Mittag-Leffler functions.Despite the inherent importance of Mittag-Leffler functions in fractional differential equations, their behaviour and types of zeros have not been fully characterized.This work resolves this delinquency by identifying regions in the domain of  and  where the Mittag-Leffler functions exhibit similar behaviour.In this work,  is restricted to real numbers.While Mittag-Leffler functions in general exhibit a diverse range of behaviors, ML functions which have the same types of zeros also exhibit many other similar properties.Hence, it is logical to organize the ML functions with similar types of zeros and similar properties into regions of the parameter space (,) (restricted to positive real numbers in this work) resulting in a - "phase diagram" for the Mittag-Leffler functions.Information extracted from a review of the literature on the theory of the zeros of ML functions together with the numerical results of this work yields the first depiction of the - phase diagram for the Mittag-Leffler functions shown in Figure 1 for the range 0 <  ≤ 3. Seven major regions or phases have been identified and descriptively labeled by an ordered pair of symbols where the first symbol of the pair indicates the number of real zeros attributed to the ML function in that phase [i.e., none (0), finite (), or infinite (∞)] and the second symbol of the pair refers to the number of complex zeros attributed to the Mittag-Leffler functions in that same phase.
Specifically, the seven phases are regions where the ML functions  , () have (1) a finite number of real zeros and an infinite number of complex zeros [/∞], (2) one real zero and an infinite number of complex zeros [1/∞], (3) an infinite number of real zeros and no complex zeros [∞/0], (4) an infinite number of real zeros and a finite number of complex zeros [∞/], and (5) a special point where  , () has no zeros at all [0/0].The remaining two phases both have no real zeros and an infinite number of complex zeros [0/∞], but exhibit different functional behaviors depending on the value of  and thus are denoted as (6) [0/∞] <1 for the region 0 <  < 1 and (7) [0/∞]  ≥ 1 for the region 1 ≤  ≤ 2. It is interesting to note that although not excluded theoretically, the phase [∞/∞] does not seem to exist in the range considered here 0 <  ≤ 3. Theoretical considerations show that the phases [0/], [/0], and [/] 2 Advances in Mathematical Physics [0/0] 0 <  < ≈ 6.669 are to be excluded.After some theoretical arguments which are applicable to the zeros of all Mittag-Leffler functions, each phase will be discussed separately with the corresponding supporting literature references.Examples will be provided for the typical behavior of the ML functions belonging to each of these phases.

Zeros of 𝐸 𝛼,𝛽 (𝑧) Theory
It follows from Hadamard's factorization theory that an entire function of fractional order has infinitely many zeros (Ahlfors [5]).The Mittag-Leffler function given by ( 1) is an entire function of order 1/ (Gorenflo and Mainardi [6]).Consequently, all ML functions  , () will have an infinite number of zeros with the possible exception of when 1/ is an integer (i.e., when  = 1, 1/2, 1/3, 1/4, . ..).Note that Hadamard's statement says nothing about the number of zeros when 1/ is an integer.In these cases, there may be no zeros, a finite number of zeros, or an infinite number of zeros.However, Sedletski [7,8] has shown that with the exception of  =  = 1 the Mittag-Leffler functions have an infinite number of zeros for  > 0. Since all terms in (1) are positive for positive , the Mittag-Leffler functions have no real zeros for  > 0. In addition, all complex zeros occur as pairs of complex conjugates (Gorenflo et al. [9]).A systematic description of the various phases follows.

Phase [0/0]
The ML function  1,1 () is equivalent to the exponential function   and is the only Mittag-Leffler function which has no zeros (Sedletski [7]) and thus is just a single point in the phase diagram.The fact that  1,1 () does not have an infinite number of zeros is consistent with Hadamard's statement since 1/ is an integer.

Phase [1/∞]
This phase extends from 0 <  < 1 and 0 <  <  with different behaviors on the boundaries as follows: the ML functions whose parameters lie on the vertical dotted line at  = 1 which separates the [1/∞] phase from [/∞] belong to the [/∞] phase (with the single exception of  1,1/2 () which belongs to [1/∞]-see Section 8 for details), while the ML functions whose parameters lie on the dotted line  =  belong to the [0/∞] phase.Figure 2 shows a typical Mittag-Leffler function in this phase.
The function is positive for positive , and as  → −∞, the function crosses the negative -axis only once and asymptotically approaches zero from below.This is evident from the asymptotic expansion of  , () given by Podlubny [1]: In this phase 0 <  < 1 and 0 <  < , thus −1 < − < 0 and Γ( − ) is negative and as  → −∞,  , () is negative and thus approaches zero from below the negative -axis.Theory confirms the fact that each  , () in this phase has only one real negative simple zero (Popov and Sedletski [10]).The fact that each ML function (except for  1,1 ()) has an infinite number of zeros for  > 0 requires that each ML function in this phase has an infinite number of complex zeros since each function has only one real zero.

Phase [0/∞] 𝛼<1
This phase extends for  ≥ , from 0 <  < 1 with the line  =  separating the phase [1/∞] from [0/∞] <1 , and Mittag-Leffler functions  , () on the line belong to the phase [0/∞] <1 .The reason ML functions on the vertical line  = 1 are not in this phase will be discussed in detail in the next section.Figure 3 shows the graph of a typical Mittag-Leffler function in the phase [0/∞] <1 .The function is positive for positive  and remains positive while asymptotically approaching the negative real axis as  → −∞ and thus has no real zeros.This is evident by applying (2) with  > , Γ( − ) is then positive and as  → −∞  , () is positive and thus approaches zero from above the negative real axis.For  = , the asymptotic expansion of  , () as  → −∞ is given by Podlubny [1]: For 0 <  < 1, Γ(−) is negative, and as  → −∞  , () is positive and this function also approaches zero from above the negative real axis.The behaviour of the Mittag-Leffler function in this phase is confirmed by the theoretical work of   Schneider [11] and Miller and Samko [12,13] who proved that for 0 <  ≤ 1 and  ≥ ,  , () is a complete monotonic nonnegative function on the negative -axis.Furthermore, Aleroev and Aleroeva [14] have proven that all zeros of  , () are complex for 0 <  < 1.Since the Mittag-Leffler functions in this phase have no real zeros, then they must have an infinite number of complex zeros.As with phase [0/∞] <1 , the Mittag-Leffler function is positive for positive  and remains positive while asymptotically approaching the negative real axis as  → −∞ and thus has no real zeros.However, the Mittag-Leffler function in phase [0/∞] ≥1 exhibits a fundamentally different functional behaviour from the Mittag-Leffler functions in phase [0/∞] <1 .This can be understood by recognizing that Mittag-Leffler functions for which  + 1 >  can be decomposed into two parts [15] given by  , (−) =  , (−) +  , (−) ,
For 1 ≤  ≤ 2, the phase [0/∞] ≥1 includes all Mittag-Leffler functions above the phase boundary line but not those on the line.This last fact will be discussed in detail in the following section.Mittag-Leffler functions  2⋅ () on the vertical boundary line  = 2 separating phase [0/∞] ≥1 from [∞/] for  > 3 belong to phase [0/∞] ≥1 as indicated by the arrow in Figure 1.This is confirmed by the theoretical work of Popov and Sedletski [10] who show that  2, () has no real zeros for  > 3.With no real zeros, each ML function in this phase must have an infinite number of complex zeros.

Phase [𝑓/∞]
This phase extends from 1 ≤  < 2 and from  > 0 to the [0/∞] ≥1 phase boundary line in Figure 1. Figure 6 shows the graph of a typical Mittag-Leffler function belonging to this phase.
The function is positive for positive  and as  → −∞, the function oscillates crossing the negative -axis a finite number of times before asymptotically approaching zero.This general behavior applies to all Mittag-Leffler functions in this phase.However, the Mittag-Leffler functions within this phase differ with respect to how they asymptotically approach zero.Some approach from above the negative  axis while others approach from below.Those Mittag-Leffler functions that approach from below must have crossed the -axis an odd number of times.For example,  ,1 () approaches from below for 1 <  < 2 and thus has an odd number of negative real zeros (Wiman [16], Gorenflo and Mainardi [15]).This is evident from (2) since  = 1 and Γ(1 − ) is negative for 1 <  < 2, therefore  ,1 () is negative as  → −∞.Similarly,  , () for 1 <  < 2 approaches zero from below and has on odd number of negative real zeros.This is evident from (3) since Γ(−) is positive for 1 <  < 2 and thus  , () is negative as  → −∞.Those Mittag-Leffler functions that approach zero from above the negative real axis must have crossed the -axis an even number of times.For example,  ,2 () for 1 <  < 2 approaches zero from above.This is evident from (2) since  = 2 and Γ(2 − ) is positive for 1 <  < 2, therefore  ,2 () is positive as  → −∞ and has an even number of negative real zeros.With only a finite number of real zeros, each ML function in this phase must have an infinite number of complex zeros.
Consequently, the phase [/∞] can be subdivided into regions where the Mittag-Leffler function has either an even or an odd number of real zeros.There are two regions where the Mittag-Leffler function has an even number of real zeros, and these two regions will be referred to as [(even)/∞].There is one region where the Mittag-Leffler

Region [𝑓(odd)/∞]
This region, which extends from 1 ≤  < 2, is bounded below by the line  =  − 1 and is bounded above by the line  =  as shown in Figure 8. Depicted in Figure 8

Region [𝑓(even)/∞]
Region [(even)/∞] appears both above and below region [(odd)/∞] on the phase diagram as shown in Figure 7.Although separated in phase space, the two regions of [(even)/∞] are fundamentally the same in that the Mittag-Leffler functions in both regions have similar behaviour.The lower region, which extends from 1 <  < 2, is bounded below by the line  = 0 and is bounded above by the line  =  − 1 as shown in Figure 9. Depicted in Figure 9 3(e).As shown in Figure 10, the subphase [2/∞] is immediately adjacent to the boundary line separating the [/∞] from the [0/∞] >1 phase.The Mittag-Leffler function in subphase [2/∞] is positive for positive  and as  → −∞ the function crosses the negative -axis reaching a minimum value below the -axis and then crosses the negative axis a second time and asymptotically approaches zero from above the negative -axis as  → −∞.For a fixed , increasing  raises the minimum value upward toward the axis causing the separation between the two zeros to decrease until the minimum value is on the -axis and the two zeros have coalesced into one real zero with a double multiplicity.This is illustrated in Figure 11  A cubic equation was found to be an approximate fit to the boundary line for 1 ≤  ≤ 3.  ≈ 0.63287 − 0.43869 + 0.79965 2 + 0.00574 3 .(8) The maximum deviation of the value of  from Table 1 compared to that calculated from ( 8) is ±0.00153 at  = 1.10; however, for  ≥ 1.85 the maximum deviation is ±0.0002.Although the points (, ) on the phase boundary have been given to 6 significant digits in Table 1, they can be determined to much greater accuracy.In particular, for the special cases whenever  and/or  are integer, these more accurately determined values can be used as reference points on the boundary and are given in Table 2.
Although the curve extends to higher , it is limited to  = 3 in this investigation.With the exception of  = 1 and  = 2, the points which constitute the boundary line are indeterminate.For example, for  = 1.5991153,  ,2 () has two real zeros and an infinite number of complex zeros and belongs to the phase [/∞].Whereas for  = 1.5991152,  ,2 () has no real zeros and an infinite number of complex zeros and belongs to the phase [0/∞].Consequently, the point on the boundary line at  = 2 which separates the phases [/∞] and [0/∞] is in the range 1.5991152 <  < 1.5991153.Although this range can be made narrower   [18] in the first attempt at an alpha-beta phase diagram.This present work is more complete including a phase not originally present and includes all the subphases in [/∞].The recent work of Duan et al. [19] shows this phase boundary line accurately for 0 ≤  ≤ 1 and approximately for  > 1 but divides the entire - phase space into only two phases and the description of his F phase having no real zeros is incorrect.

Phase [∞/0]
This phase extends from 2 ≤  ≤ 3 and from  > 0 to the boundary line shown in Figure 1. Figure 12 shows the graph of a typical Mittag-Leffler function in this phase.The function is positive for positive , and as  → −∞, the function oscillates with increasing amplitude crossing the negative axis an infinite number of times.
Nevertheless it is still true that the zeros of  2,3 () are negative and real.For  > 3, it has been shown (Popov and Sedletski [10]) that  2, () has no real zeros and thus belongs to phase [0, ∞].Thus Mittag-Leffler functions on the line  = 2 for 0 <  ≤ 3 belong to the phase [0/∞] as indicated by the arrow in Figure 1.
There is considerable theoretical work searching for all pairs (, ) such that the zeros of the Mittag-Leffler functions are negative, real, and simple (i.e., Popov [23][24][25][26], Popov and Sedletski [10], and Ostrovskii and Peresyolkova [22]).For example, Popov [25] showed that for  > 2 and 0 <  ≤ (2− 1) all zeros of  , () are negative, real, and simple.Although this covers the majority of phase [∞/0] the upper limit on  is more restrictive than (8) predicts.However, the work of Popov and others is also more restrictive requiring the zeros to be simple.The phases proposed in Figure 1 are only concerned with whether the zeros are real or complex.The general properties attributed to the Mittag-Leffler functions in each phase hold true regardless of whether the zeros are simple or multiples.
Although the phase diagram in Figure 1 is restricted to 0 <  ≤ 3, it is conjectured that phase [0/∞] extends to 2 ≤  ≤ ∞ based on the theoretical work of Popov and the others listed above.Also supporting this hypothesis is the work of Wiman [16], Pólya [27], and Ostrovskii and Peresyokkova [22] who have shown that all zeros of  ,1 () for  ≥ 2 are real, negative, and simple.In addition, Popov [26] has shown that all zeros of the function  ,+1 () are real, negative, and simple for  ≥ 3 and that all zeros of  4,9 () are real, negative, and simple.

Phase [∞/𝑓]
This phase extends from 2 <  ≤ 3 and for  above the phase boundary line given approximately by (8). Figure 13   The function is positive for positive  and as  → −∞ the function oscillates with growing amplitude crossing the negative real axis an infinite number of times.The Mittag-Leffler functions in phase [∞/] differ from those in phase [∞/0] in that the functions in phase [∞/] exhibit a finite number of oscillations whose relative minima are above the negative -axis, whereas for functions in phase [∞/0], the relative minima for all oscillations are below the negative -axis (see Figure 12).For fixed , and  incrementally above the phase boundary line, the relative minimum in the oscillation nearest  = 0 no longer lies below the negative axis but is now above the negative -axis and thus the function has lost two real zeros and gained two complex zeros (one and its complex conjugate).The phase boundary occurs at the value of  when this relative minimum rests on the negative -axis.As  is increased further, the relative minimum of the second oscillation nearest  = 0 also occurs above the negative -axis (shown in Figure 13) admitting two more complex zeros for a total of 4 complex zeros and an infinite number of real zeros.This process continues as  is increased; each time the relative minimum of an oscillation moves above the negative -axis the function loses two real zeros and gains two complex zeros.This phase has been extensively mapped by the authors and is shown in Figure 14.Reference values on the subphase boundary lines are given in Table 4.
Note that all Mittag-Leffler functions in this phase just above the boundary line have 2 complex zeros and thus this region is listed as [∞/2].Since each of these regions [∞/2], [∞/4], [∞/6], . . .all has the same general behavior, they are considered as one major phase [∞/] with an infinite number of subphases.An interesting observation from Figure 14 is that the phase [∞/∞] can never be achieved but is only approached for  > 3 and  = 2 +  as  becomes incrementally small.

Summary
The depiction of the - phase diagram for the Mittag-Leffler functions with real arguments is represented by Figure 1 with subphases shown in more detail in Figures 7,8,9,10,and 14.Each phase represents a region where  , () have not only the same type of zeros but also exhibit similar functional behavior.This represents a major step forward in characterizing the nature of the zeros and organizing the Mittag-Leffler functions according to their behavior.For complex arguments, the reader is referred to the numerical calculations for  , () in the complex plane by Hilfer and Seybold [28].Their results are consistent with the present work where the efforts overlap.

Figure 1 :
Figure 1: - phase diagram for  , ().The arrows point to the particular phase into which a Mittag-Leffler function whose parameters lie on a phase boundary belongs.Dotted lines are used to indicate phase boundaries where  and  are known exactly and solid lines for boundaries where  and/or  may be determined as accurately as desired but not known exactly.In opposition to the remainder of the points on the line  = 1, the dark points at  = 3/2, 5/2, 7/2, 9/2, . . .belong to phase [0/∞] <1 and  = 1/2 to phase [1/∞].
are the lines separating the various subphases [/∞] where the Mittag-Leffler function has  = 1, 3, 5, 7, . . .zeros.Lines begin on the line  =  curl around and end at the point  =  = 1.Each subsequent line is asymptotically closer to the  = −1 line as it approaches the point  =  = 1.The number of subphases drawn in Figure 8 was determined by when there was no clear
are the lines separating the various subphases [/∞] where the Mittag-Leffler function has  = 2, 4, 6, 8, . . .zeros.Lines begin on the line  =  − 1 curl around and end on the line  = 0.The number of subphases drawn in Figure 7 was determined by when there was no clear separation distinguishing consecutive subphases.The line separating subphase [100/∞] from [102/∞] is also drawn for perspective.Points on the boundary lines separating consecutive subphases can be determined numerically as accurately as desired, but no exact values are known.Two points on each curve will be given as reference values which are accurate to 11 significant digits.Points on the line  =  − 1 where the boundary line separates subphase [/∞] from subphase [( + 2)/∞] for  = 2, 4, 6, . . .102 are given in Table 3(c).Additional reference values are given in Table 3(d) which lists points on the horizontal line  = 0 where the boundary line separates subphase [/∞] from subphase [(+ 2)/∞] for  = 2, 4, 6, . . .102.Note that the tabulated values in Tables 3(a) and 3(d) are related.This is because  ,0 () =  , () and consequently all zeros of  , () are also zeros of  ,0 () except that  ,0 () has an additional zero at  = 0. Thus, whereas  , () has an odd number of zeros, the added zero gives  ,0 () an even number of zeros.The other portion of the phase diagram in the phase [(even)/∞] extends from 1 <  < 2 and is bounded below by the line  =  and bounded above by the [/∞] to [0/∞] phase boundary line as shown in Figure 10.Depicted in Figure 10 are the lines separating the various subphases [/∞] where the Mittag-Leffler function has  = 2, 4, 6, . . .zeros.The boundary lines begin at the point  = 2,  = 3 and end at the point  =  = 1.Each subsequent line is asymptotically closer to the  =  line as it approaches the point  =  = 1.The number of subphases drawn in Figure 8 was determined by when there was no clear separation distinguishing consecutive subphases.The line separating subphase [100/∞] from [102/∞] is also drawn for perspective.Points on the boundary lines separating the consecutive subphases can be determined numerically as accurately as desired, but no exact values are known other than the two end points.An additional point on each curve
≥1 as discussed earlier.The value of  above which the function has no real zeros delineates the boundary between the phases [/∞] and [0/∞] for 1 <  < 2, and Table1lists these approximate values for various values of  (see the discussion of phase [∞/] for the methodology in the determination of the  values for 2 <  ≤ 3).
with the Mittag-Leffler function  , () with  = 7/4.For  just less than 2.34513, the function has two zeros centered around  = −20 and increasing  to ≈2.34513 results in one real zero with a double multiplicity.A further increase in  yields a function with no real zeros which oscillates as it asymptotically approaches  = −∞ which is characteristic of a function in phase [0/∞]

Table 1 :
Boundary line data points.
(with the exception of the two points known exactly:  =  = 1 and  = 2,  = 3).It must be noted that all approximate values of  in Tables1 and 2correspond to Mittag-Leffler functions in phase [/∞] for 1 <  < 2 and in phase [∞/0] for 2 ≤  ≤ 3 and by adding one digit to the least significant digit of  the function appears on the opposite side of

Table 2 :
Boundary line reference points.
the boundary line in phase [0/∞] ≥1 for 1 <  < 2 or phase [∞/] for 2 ≤  ≤ 3.While the opposite is true for the approximate values of  in Table 2 which are in either phase [0/∞] ≥1 or [∞/] initially and after adding one digit to the least significant digit of  are in phase [/∞] or [∞/0].It should be noted that this phase boundary line was numerically determined first byHanneken et al.

Table 3 :
Reference values in phase [/∞].(a) Reference value transition points along the line  = shows a typical Mittag-Leffler function in this phase.