Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform

Le Roy , Lerch and Le gendre chi functions and generalised Borel-Le Roy transform Giuseppe Dattoli * 1 and Roberto Ricci † 1 1 ENEA, Nuclear Department, F rascati Resear ch Center , V ia E. F ermi 45, 00044 F r ascati (Rome), Italy Abstract The Le Roy function has been the focus of intensi ve research in recent years, o w- ing both to its rele v ance in analysis and its versatility in applications in volving fractional di ff erential operators. Other special functions – such as the Lerch tran- scendent and the Legendre chi function – ha ve found applications ranging from Bose-Einstein and Fermi-Dirac statistics in physics to pure mathematical in vesti- gations in volving polylogarithms and Dirichlet L-series. In this article, we present a unified frame work based on a recent reformulation of Indicial Umbral Theory (IUT) grounded in the formal theory of po wer series. W ithin this setting, we study the properties and generalisations of these special functions. In particular , we b uild upon the revised formulation of IUT to incor- porate the role of the Borel-Le Ro y transform, and to e xplore the extension of the formalism to di ver gent series via appropriate resummation techniques. K eywords— indicial umbral theory , special functions, Borel-Le Roy transform Intr oduction The theory of special functions has undergone a series of ev olutionary stages. It ini- tially arose from the study of the analytic properties of certain functions and polynomials deemed ”special”, and gradually developed into a pursuit of a unified conceptual frame- work. Authoritati ve textbooks o ff er a synthesis of these de velopments (for a partial list, * E-mail: pinodattoli@libero.it (Giuseppe Dattoli) † E-mail: roberto.ricci@enea.it (Roberto Ricci) 1 see refs. [ 1 , 2 , 3 , 4 , 5 , 6 ]), which emerged within specific conte xts shaped by the prev ailing mathematical paradigms of their time. A central challenge in this evolution has been identifying the property that qualifies a function as ”special”. A pi votal moment in this discourse was marked by W igner’ s perspecti ve, articulated in his Princeton lectures, which catalysed a paradigm shift: the reinterpretation of special functions as matrix elements of Lie group representations [ 7 , 8 ]. This group-theoretic formulation opened a ne w path for unification. The work of T alman [ 9 ] represented a significant advance in this direction, o ff ering a seminal contribution that pa ved the way for further in vestigations [ 10 , 11 ]. It became increasingly clear that special functions are specific solutions to families of ordinary di ff erential equations (ODEs) with non-constant coe ffi cients [ 12 ]. This real- isation provided the foundation for alternati ve unifying strategies. In particular , the hy- pergeometric di ff erential equation and its associated solutions ha ve emer ged as especially well-suited to this goal. An additional unifying frame work has been o ff ered by the Umbral Calculus (UC), whose conceptual roots span nearly two centuries [ 13 ]. The theory originated from the observ ation of formal analogies among seemingly unrelated special polynomials, which – under suitable conditions – could be treated as ordinary monomials [ 14 ]. This insight culminated in Ste ff ensen’ s introduction of po weroids [ 15 ], and later in the development of the Quasi-Monomiality formalism by Dattoli and T orre [ 16 ]. In the 1970s, Roman and Rota provided a rigorous foundation for the Umbral Cal- culus by applying the theory of linear functionals. The y demonstrated that UC could be formalised as an algebra generated by linear functionals on the vector space of polynomi- als in a v ariable z [ 17 ]. More recently , UC has e volv ed into the Indicial Umbral Theory (IUT) [ 18 , 19 ], which incorporates special functions and introduces e ff ective computational tools through a re- definition of umbral techniques and a refined use of Borel-Laplace transforms [ 19 , 20 ]. In particular , ref. [ 19 ] presents a reformulation of the original IUT (initially termed Indicial Umbral Calculus, IUC), grounded in the di ff erential algebra of formal po wer series [ 21 ]. This frame work enables a mathematically sound definition of the umbral operator – a perspecti ve reminiscent of Roman and Rota’ s algebraic approach, but with a significantly greater emphasis on analytic properties. A notable byproduct of this formulation is its ability to extend the umbral formalism to di ver gent formal power series, interpreting them as asymptotic expansions that can be resummed via generalised versions of the Borel-Laplace transform. In this article, we apply IUT techniques to the study of the Le Roy , Lerch, and Leg- endre functions. W e demonstrate that these functions serv e as valuable benchmarks for testing the potential of the method and uncov ering ne w structural properties of the func- tions themselves. 2 1 Le Roy function The function defined by the series (1) L ( ζ ; µ ) · · = ∞ X r = 0 ζ r Γ (1 + r ) µ , µ ∈ R . was introduced by Le Roy at the be ginning of the last century [ 22 ] and employed in studies regarding the asymptotic properties of the analytic continuation of the sum of po wer series. More recently , sev eral authoritativ e articles (for a partial list see [ 23 , 24 , 25 , 26 , 27 , 28 ] and references therein) have renewed the interest in this function, by establishing generalisations, opening new fields of research – e.g. in fractional calculus – and propos- ing novel applications. As a significativ e example, in a paper dedicated to the study of stochastic di ff erential equations, K olokoltsov [ 29 ] considered the function L ( ζ ; 1 / 2), stating that this function plays the same role for stochastic equations as the ordinary ex- ponential for the deterministic case. In order to in vestigate the properties of the Le Roy function in the context of the indicial umbral theory (IUT) [ 18 , 30 ], we introduce the follo wing class of ground states: (2) ϕ µ α, β ( t ) · · = 1 Γ ( β + α t ) µ , α > 0 , β ∈ C , µ ∈ R , which is an obvious generalisation of the commonly used class: (3) ϕ α, β ( t ) · · = 1 Γ ( β + α t ) , α > 0 , β ∈ C , µ ∈ R . W e also consider the specialisation of eq. ( 2 ) obtained by setting α = β = 1, i.e. the particular class of ground states: (4) ϕ µ ( t ) · · = ϕ µ 1 , 1 ( t ) = 1 Γ (1 + t ) µ , µ ∈ R . By using eq. ( 4 ), it is possible to express eq. ( 1 ) in the follo wing simple umbral form: (5) L ( ζ ; µ ) = 1 1 − ζ u [ ϕ µ ] = e ζ u [ ϕ µ − 1 ] . On the other hand, by applying the same umbral operator to the general ground state eq. ( 2 ), we obtain one of the possible generalisations of the Le Roy function [ 31 ]. Namely: (6) L ( ζ ; α, β, µ ) = e ζ u [ ϕ µ − 1 α, β ] · · = ∞ X r = 0 ζ r r ! Γ (1 + β + α r ) µ − 1 . Both umbral identity eq. ( 5 ) and eq. ( 6 ) are easily prov ed by expanding the exponential in Maclaurin series and exploiting the definition of u as a functional in the di ff erential subalgebra of analytically con verging formal series [ 19 ], namely: (7) u r [ φ ] · · = φ ( r ), φ ∈ C { t } ⊂ C ⟦ t ⟧ , 3 provided r is within the domain of con vergence of φ . F or further comments see [ 18 , 30 , 31 , 32 ]. The use of the pre vious umbral restyling enables a significant simplification of the study of the properties of Le Roy function and its generalisations. T aking e.g. the repeated deri vati ve with respect to ζ of both sides of eq. ( 5 ), one e ventually finds: (8) ∂ n ζ L ( ζ ; µ ) = u n e ζ u [ ϕ µ − 1 ] = ∞ X r = 0 ζ r r ! Γ (1 + n + r ) µ − 1 = L ( ζ ; 1 , n , µ ) , where use has been made of the definition ( 6 ). IUT methods also enable the e valuation of infinite inte grals in volving the K olokoltso v function (for further comments see [ 18 , 30 ]), for example: (9) Z ∞ −∞ d ζ L ( − ζ 2 ; 1 2 ) = Z ∞ −∞ d ζ e − ζ 2 u [ ϕ − 1 / 2 ] = √ π u − 1 / 2 [ ϕ − 1 / 2 ] = 4 p π 3 . The same formalism can be employed to study the three-parameter Mittag-Le ffl er a.k.a. Prabhakar function [ 33 ]: (10) E α, β, γ ( ζ ) = ∞ X r = 0 ( γ ) r r ! Γ ( α r + β ) ζ r , α > 0 , β, γ ∈ C . Using the ground state ψ α, β ,γ ( t ) = ( γ ) t Γ ( α t + β ) ≡ Γ ( γ + t ) Γ ( γ ) Γ ( α t + β ) , α > 0 , β, γ ∈ C , where ( γ ) t · · = Γ ( γ + t ) / Γ ( γ ) is a generalised version of the rising Pochhammer symbol, eq. ( 10 ) can be expressed in the umbral form: (11) E α, β, γ ( ζ ) = e ζ u [ ψ α, β, γ ] . An example of the utility of the IUT formalism is provided by the e v aluation of the successi ve deriv atives of eq. ( 10 ). Applying the ∂ n ζ operator to both sides of eq. ( 11 ) and proceeding as before we obtain: (12) ∂ n ζ E α, β, γ ( ζ ) = u n e ζ u [ ψ α, β, γ ] = ∞ X r = 0 ζ r u r + n r ! [ ψ α, β, γ ] = ∞ X r = 0 Γ ( γ + r + n ) Γ ( γ ) Γ ( α r + β + α n ) ζ r r ! = ( γ ) n E α, β + α n , γ + n ( ζ ) . Another example of application in volving integration is (see [ 34 ] for a discussion on Pochhammer symbols with negati ve index): (13) Z ∞ −∞ d ζ E α, β, γ ( − ζ 2 ) = Z ∞ −∞ d ζ e − ζ 2 u [ ψ α, β, γ ] = √ π u − 1 / 2 [ ψ α, β, γ ] = √ π ( γ ) − 1 / 2 Γ ( β − α/ 2) . 4 Before concluding this opening section, we introduce a further element of discussion, whose role is emphasised in the final part of the article. Let us consider the inte gral Bor el transform of the Leroy function, defined as: (14) L ( µ ) B ( ζ ) ≡ B [ L ( µ ) ]( ζ ) · · = Z ∞ 0 d t e − t L ( µ ) ( ζ t ) , where we hav e adopted for conv enience the alternative notation L ( µ ) ( ζ ) ≡ L ( ζ ; µ ). Note that (15) Z ∞ 0 d t e − t L ( µ ) ( ζ t ) = 1 ζ Z ∞ 0 d x e − x /ζ L ( µ ) ( x ) ≡ 1 ζ L 1 [ L ( µ ) ]( ζ ) , i.e. ζ L ( µ ) B ( ζ ) coincides with the generalised Laplace transform of order 1 of the function L ( µ ) . On the other hand, as a power series, (16) L ( µ ) ( ζ ) = B 1 [ ˜ L ( µ − 1) ]( ζ ) , where ˜ L ( µ ) ( ζ ) · · = ζ L ( µ ) ( ζ ) and B 1 is the formal Bor el transform operator of order 1 [ 19 ]. Since the series ˜ L ( µ − 1) con verges to an entire function, we kno w from the theory of Borel- Laplace resummation that L 1 [B 1 [ ˜ L ( µ − 1) ]]( ζ ) = ˜ L ( µ − 1) ( ζ ). It follows that: (17) L ( µ ) B ( ζ ) = L ( µ − 1) ( ζ ) . Equation ( 17 ) can be easily verified by directly solving the inte gral in eq. ( 14 ). A similar argument shows that the following Borel-Le Ro y transform of the gener- alised Le Roy function eq. ( 6 ): (18) L BL ( ζ ; α, β, µ ) · · = Z ∞ 0 d t e − t t µ L ( ζ t β ; α, β, µ ) , satisfies the identity: (19) L BL ( ζ ; α, β, µ ) = L ( ζ ; α, β, µ − 1) . Further examples will be discussed in the section de voted to final comments. In this introductory section we have touched on the formalism we will use in the forthcoming part of the article, where we frame in the IUT context the Lerch transcendent and Legendre chi function. 2 Ler ch transcendent The Lerch transcendent [ 35 ] is defined by the series: (20) Φ ( ζ ; α, s ) · · = ∞ X r = 0 ζ r ( r + α ) s , con verging for an y α > 0 in | ζ | < 1, and also in | ζ | = 1 if Re s > 0 . 5 The reason for the interest in this function stems from its association with the polylog- arithm function [ 36 ], Dirichlet η and Riemann-Hurwitz ζ functions [ 37 ]. By introducing the ne w classes of ground states (21a) ν α, β, s ( t ) · · = ( β ) t ( t + α ) s , (21b) ν α, s ( t ) · · = ν α, 1 , s ( t ) = Γ (1 + t ) ( t + α ) s , it is e vident that eq. ( 20 ) can be written in umbral form as: (22) Φ ( ζ ; α, s ) = e ζ u [ ν α, s ] . By applying the same umbral operator to the general ground state eq. ( 21a ), we obtain the generalised Lerch transcendent: (23) Φ ( ζ ; α, β, s ) · · = ∞ X r = 0 ( β ) r ( r + α ) s ζ r r ! = e ζ u [ ν α, β, s ] , which reduces to eq. ( 22 ) for β = 1. If we are interested to the properties under deri v ativ e of these functions, we can take again adv antage from their exponential umbral images and find: (24a) ∂ n ζ Φ ( ζ ; α, s ) = u n e ζ u [ ν α, s ] = ∞ X r = 0 Γ (1 + r + n ) ( r + n + α ) s ζ r r ! = (1) n Φ ( ζ ; α + n , 1 + n , s ) , (24b) ∂ n ζ Φ ( ζ ; α, β, s ) = u n e ζ u [ ν α, β, s ] = ( β ) n Φ ( ζ ; α + n , β + n , s ) . W e ha ve already mentioned the importance of the Lerch transcendent function, de- scending from the fact that man y special functions can be defined through it or can be deri ved as a particular case. F or example, the in verse tangent integral, denoted by the symbol T i( ζ ) [ 38 ], is defined in terms of the generalised Lerch function as: (25) 2 s T i( ζ ) ζ = Φ ( − ζ 2 ; 1 2 , 1 , s ) . The umbral image of the r .h.s. of the previous identity is a Gaussian, namely: (26) 2 s T i( ζ ) ζ = e − ζ 2 u [ ν 1 2 , 1 , s ] . The successiv e deriv ativ es with respect to ζ of both sides of eq. ( 29 ) can be easily obtained exploiting the follo wing rule, valid for an ordinary Gaussian [ 18 ]: (27) ∂ n ζ e a ζ 2 = H n (2 a ζ , a ) e a ζ 2 , 6 where H n ( x , y ) are the two-v ariable Hermite-Kamp ´ e de F ´ eri ´ et polynomials: (28) H n ( x , y ) · · = n ! ⌊ n 2 ⌋ X r = 0 x n − 2 r y r ( n − 2 r )! r ! . Using eq. ( 27 ), we e ventually obtain: (29) 2 s ∂ n ζ T i( ζ ) ζ ! = H n ( − 2 ζ u , − u ) e − ζ 2 u [ ν 1 2 , 1 , s ] = ( − 1) n n ! ⌊ n 2 ⌋ X r = 0 ( − 1) r (2 ζ ) n − 2 r ( n − 2 r )! r ! u n − r e − ζ 2 u [ ν 1 2 , 1 , s ] = ( − 1) n n ! ⌊ n 2 ⌋ X r = 0 ( − 1) r (2 ζ ) n − 2 r ( n − 2 r )! r ! (1) n − r Φ ( − ζ 2 ; n − r + 1 / 2 , 1 + n − r , s ) . Before concluding this section, we would lik e to underline the link between the Lerch function and the polylogarithm function. Their entanglement is well known [ 38 ] and the generalisations we hav e discussed so far o ff er an IUT vie w of the polylogarithm function, which writes: (30) Li s ( ζ ) = ∞ X r = 0 ζ r r s = ζ Φ ( ζ ; 1 , 1 , s ) . It is e vident that (31) Li s ( ζ ) ζ = e ζ u [ ν 1 , 1 , s ] , hence the properties of polylogarithms can be studied using this exponential ν 1 , 1 , s -umbral image. It is, for example, easily checked that (32) Li s ( ζ 1 + ζ 2 ) ζ 1 + ζ 2 = e ( ζ 1 + ζ 2 ) u [ ν 1 , 1 , s ] = ∞ X r = 0 ζ r 1 r ! r ! Φ ( ζ 2 , 1 + r , 1 + r , s ) . Although we hav e so far only considered integer po wers of the umbral operator u , the expression (33) u λ [ ν α, s ] = Γ (1 + λ ) ( λ + α ) s is well defined for any complex λ , provided t = λ does not correspond to a singularity of the function ν α, s ( t ). This enables to define ”polylogarithms of non-integer order” such as: (34) g s ( ζ ) = Li (1 / 2) s ( ζ ) ζ = e ζ u 1 / 2 [ ν 1 , 1 , s ] = ∞ X r = 0 Γ (1 + r / 2) (1 + r / 2) s ζ r r ! . It is interesting to note that (35) Z ∞ −∞ d x e − x 2 g s (2 x ζ ) = √ π Li s ( ζ 2 ) ζ 2 . 7 The above identity , whose proof is sketched in appendix A.1 , states that the polylogarithm function is the Gauss transform of its counterpart of order 1 / 2. W e ha ve underscored that that the defining series of the Lerch transcendent has a lim- ited interv al of conv ergence. W e can ho wev er extend the domain of the function by ana- lytic continuation. In particular , the following integral representation (see appendix A.2 ): (36) Φ ( ζ , α, s ) = 1 Γ ( s ) Z ∞ 0 d t t s − 1 x e − α t 1 − ζ e − t , reproduces the series eq. ( 20 ) for | ζ | < 1, but con verges for v alues of the variable exceed- ing unity , namely ζ ∈ C \ [1 , ∞ ) , Re s > 0 , Re α > 0 (see Fig. (1)). The integral formula also holds in ζ = 1 if Re s > 1. 3 Legendr e χ function The χ function, introduced by Legendre in his book Exer cice de Calcul Int ´ egral sur divers or dres de tr anscendantes et sur les quadr atur es , has been the subject of influential re- search. A partial list of more recent studies is reported in refs. [ 39 ]. In modern notation, the χ function is defined as: (37) χ s ( ζ ) · · = ∞ X r = 0 ζ 2 r + 1 (2 r + 1) s = ζ 2 s Φ ( ζ 2 ; 1 2 , s ), | ζ | < 1 , where we hav e made explicit its relationship with the Lerch transcendent. In virtue of the follo wing decomposition: (38) Φ ( ζ ; 1 , s ) = e ζ u [ ν 1 , s ] = cosh( ζ u ) [ ν 1 , s ] + sinh( ζ u ) [ ν 1 , s ] = c s ( ζ ) + s s ( ζ ) , where (39a) c s ( ζ ) · · = cosh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 ζ 2 r (2 r + 1) s , (39b) s s ( ζ ) · · = sinh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 ζ 2 r + 1 (2 r + 2) s , we obtain by direct inspection the follo wing ν 1 , s -umbral image for χ s : (40) χ s ( ζ ) = ζ c s ( ζ ) = ζ cosh( ζ u ) [ ν 1 , s ] . It is worth noting that: (41a) ∂ ζ c s ( ζ ) = u sinh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 2 r + 2 (2 r + 3) s ζ 2 r + 1 , (41b) ∂ ζ s s ( ζ ) = u cosh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 2 r + 1 (2 r + 2) s ζ 2 r . 8 Even though the use of the umbral formalism is not crucial for the deri vation of the pre vi- ous identities, it may simplify the follo wing generalisations: (42a) ∂ 2 n ζ c s ( ζ ) = u 2 n sinh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 (2 n + 1) 2 r (2 r + 2 n + 1) s ζ 2 r , (42b) ∂ 2 n ζ s s ( ζ ) = u 2 n cosh( ζ u ) [ ν 1 , s ] = ∞ X r = 0 (2 n + 1) 2 r + 1 (2 r + 2 n + 2) s ζ 2 r + 1 . In analogy to the case of Lerch transcendent, the domain of definition of the Legendre chi function can be e xtended to a lar ger region using the following integral representation (see appendix A.2 ): (43) χ s ( ζ ) = ζ Γ ( s ) Z ∞ 0 d t t s − 1 e − t 1 − ζ 2 e − 2 t . 4 P olygamma function The polygamma function of order m ( m -polygamma for short) is a meromorphic function on the complex plain, defined as the ( m + 1)-th deriv ati ve of the logarithm of the Gamma function: (44) ψ ( m ) ( α ) · · = ∂ m α ψ ( α ) = ∂ m + 1 α ln Γ ( α ), m ∈ N 0 , where ψ (0) ( α ) = ψ ( α ) = Γ ′ ( α ) / Γ ( α ) is the digamma function. The m -polygamma is holomorphic on C \ Z ≤ 0 , with poles of order m + 1 at all the nonpositiv e integers. Its relation with the Lerch transcendental is provided by the formula: (45) ψ ( m ) ( α ) = ( − 1) m + 1 m ! Φ (1; α, m + 1) . It immediately follo ws from eq. ( 22 ) that: (46) ψ ( m ) ( α ) = ( − 1) m + 1 m ! e u [ ν α, m + 1 ] = ( − 1) m + 1 m ! ∞ X r = 0 1 ( r + α ) m + 1 = ( − 1) m + 1 m ! ζ ( m + 1 , α ) , where ζ ( s , α ) = Φ (1; α, s ) = e u [ ν α, m + 1 ] is the Hurwitz zeta function. By using in eq. ( 45 ) the identity (47) m ! = Γ ( m + 1) = Z ∞ 0 d x x m e − x , after the change of integration variable x 7→ t ( r + α ) and the interchange of sum and integral we obtain the inte gral representation: (48) ψ ( m ) ( α ) = ( − 1) m + 1 Z ∞ 0 d t t m e − α t 1 − e − t , con verging for m > 0 and Re α > 0. Further comments on the relev ance and importance of the polygamma function for IUT will be discussed in the forthcoming conclusi ve section. 9 5 Conclusions One of the leitmoti vs of the present in vestigation has been the possibility of extending the con ver gence of the series representativ e of the functions we ha ve introduced, using di ff erent forms of the integral representation. This is a noticeable element of discussion because of its potential application in the context of summability . As is well known the Euler series [ 40 , 41 ] (49) d 1 ( x ) · · = ∞ X r = 0 ( − 1) r r ! x r , emerging from the perturbati ve solution of the equation (50) x 2 y ′ + y = x , y (0) = 0 , has zero con vergence radius. Ho we ver , using the integral representation of the factorial, the series in eq. ( 49 ) can be cast in the form of the integral representation, reported belo w: (51) d 1 ( x ) = ∞ X r = 0 ( − 1 x ) r Z ∞ 0 d t e − t t r ≈ Z ∞ 0 d t e − t 1 + xt = ¯ d 1 ( x ) , obtained after interchanging the symbols of summation and integral. This is an abuse (hence the symbol ≈ , instead of the ordinary symbol of equality), since it holds for v alues of the variable x allo wing the conv ergence of the series in eq. ( 51 ). The consequence of this illegitimate procedure is tw ofold: 1. d 1 ( x ) has been associated with an integral transform con ver ging for all positiv e x v alues 2. The solution of the di ff erential equation in eq. ( 50 ) can be written as (52) y ( x ) = x ¯ d 1 ( x ) . A further example of a similar manipulation is o ff ered by the identities: (53) d 2 ( x ) · · = ∞ X r = 0 ( − 1) r ( r ! ) 2 x r ≈ Z ∞ 0 d u Z ∞ 0 d v e − u + v 1 + uv x = ¯ d 2 ( x ) , which state that an even more div erging series can be associated with a well-beha ved function in the positiv e x region. The price to be paid is the introduction of a double integral accounting for the squared factorial. The possible extension to an y integer po wer in terms of multiple integral transforms is easily guessed. A small step further is accomplished by considering the identities: (54) d 2 ( x ; α, β ) · · = ∞ X r = 0 ( − 1) r ( Γ (1 + β + α r ) 2 x r ≈ Z ∞ 0 d u Z ∞ 0 d v e − u + v ( uv ) β 1 + ( uv ) α x = ¯ d 2 ( x ; α, β ) , 10 suggesting the possibility of an extension of the IUT analysis for inte ger order Le Roy function, namely L ( x , µ ) , µ ∈ N + . The interest for this aspect of the problem is enhanced by the fact that L ( x , n ) are Humbert type Bessel [ 42 , 43 ] and by the possibility of introducing a multidimensional Borel-Le Roy transform as e. g.: (55a) e − x = Z ∞ 0 d u Z ∞ 0 d v u β v δ L ( xu e γ v ; α, β, γ , δ ) , (55b) L ( x ; α, β, γ , δ ) = ∞ X r = 0 x r Γ (1 + β + α r ) Γ (1 + δ + γ r ) . Regarding the discussion associated to the Polygamma function, we should empha- size that its introduction seems to be extraneous to the umbral formalism outlined in this article. W e did not use an y umbra v acuum and image function to define their framing within the IUT framework. This can howe ver be easily fixed, if we note that we can reinterpret the integral representation eq. ( 48 ) as a special case of (56) ψ ( m ) ( α, ζ ) = − ∂ m α Z ∞ 0 d t e − α t 1 − ζ e − t , which realizes a two variable Polygamma and reduces to the Lerch function. Indeed we find the obvious generalization of eq. ( 45 ) (57) ψ ( m ) ( α, ζ ) = Γ ( m + 1) Φ ( ζ ; α, m + 1) . The translation of eq. ( 56 ) in umbral terms is straightforward and will be discussed else- where, within a more general context. These final comments yield an idea of the directions along which future researches can be de veloped and will be discussed by the present authors. A A ppendix A.1 Deriv ation of eq. ( 35 ) In order to deri ve eq. ( 35 ), we e xploit the umbral identity (A.1) g s (2 x ζ ) = e 2 x ζ u 1 / 2 [ ν 1 , 1 , s ] to write: (A.2) I ( ζ ) = Z ∞ −∞ d x e − x 2 g s (2 x ζ ) = Z ∞ −∞ d x e − x 2 + 2 x ζ u 1 / 2 [ ν 1 , 1 , s ] . By ”completing the square” we easily obtain (A.3) I ( ζ ) = Z ∞ −∞ d y e − y 2 e ζ 2 u [ ν 1 , 1 , s ] = √ π Li s ( ζ 2 ) ζ 2 , where y · · = x − ζ u 1 / 2 and use has been made of the umbral identity: (A.4) e ζ 2 u [ ν 1 , 1 , s ] = Li s ( ζ 2 ) ζ 2 . 11 A.2 Deriv ation of eq. ( 36 ) and eq. ( 43 ) In order to prove eq. ( 36 ), we use the inte gral definition of the g amma function and write: (A.5) Φ ( ζ , α, s ) Γ ( s ) = ∞ X r = 0 ζ r ( r + α ) s Z ∞ 0 d x x x s e − x . By changing the integration variable, x → t = x / ( n + α ), and after interchanging the sum and integral, we obtain: (A.6) Φ ( ζ , α, s ) Γ ( s ) = ∞ X r = 0 Z ∞ 0 d t t t s ζ r e − ( r + α ) t = Z ∞ 0 d t t t s e − α t ∞ X r = 0 ( ζ e − t ) r = Z ∞ 0 d t t s − 1 e − α t 1 − ζ e − t . The deri vation of eq. ( 43 ) is completely analogous. Refer ences [1] R. Courant and D. Hilbert. Methods of Mathematical Physics , v olume I. Interscience Publishers, 1953. [2] P . M. Morse and H. Feshbach. 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