A note on Fox's H function in the light of Braaksma's results
aa r X i v : . [ m a t h . C V ] A p r A note on Fox’s H function in the light of Braaksma’s results D.B. Karp a , b ∗ a Far Eastern Federal University, Vladivostok, Russia b Institute of Applied Mathematics, FEBRAS, Vladivostok, Russia
Abstract
In our previous works we found a power series expansion of a particular case of Fox’s H function H q, p,q in a neighborhood of its positive singularity. An inverse factorial series expansionof the integrand of H q, p,q served as our main tool. However, a necessary restriction on parametersis missing in those works. In this note we fill this gap and give a simpler and shorter proof of theexpansion around the positive singular point. We further identify more precisely the abscissaof convergence of the underlying inverse factorial series. Our new proof hinges on a slightgeneralization of a particular case of Braaksma’s theorem about analytic continuation of Fox’s H function. Keywords:
Fox’s H function, gamma function, inverse factorial series, Nørlund-Bernoulli poly-nomial MSC2010: 33C60, 33B15, 30B50, 11B68
Fox’s H function is a very general function defined by the Mellin-Barnes integral (1) below. Itwas introduced by Fox [4] in the context of symmetrical Fourier kernels and has been studiedby a number of authors thereafter [1, 9, 11, 14]. Fox’s H function has a number of importantapplications, most notably in statistics [2, 5, 11] and fractional calculus [10, 11]. Braaksma’smammoth manuscript [1] remains among the deepest investigations on the topic. In this note wewill be only interested in the case when the parameter µ defined in (2) equals zero (Braaksma’swork contains a careful study of both cases µ > µ = 0, while the case µ < µ > µ = 0 the integral (1) defining the H functiononly converges if | z | < β − , where β is given in (2). It also converges for | z | > β − but overa different integration contour so that the two functions obtained in this way are not, generallyspeaking, analytic continuations of each other. Braaksma constructed analytic continuations of the H function to an infinite set of sectors forming a partition of the Riemann surface of the logarithm.The points of intersection of the circle | z | = β − with the rays bounding these sectors are singularpoints of H . He further derived analytic continuation from each such sector to the domain | z | > β − on this Riemann surface.Under certain restrictions to be specified below the positive semi-axis arg( z ) = 0 forms theboundary between two such sectors, so that the point z = β − is a singular point of Fox’s H function. ∗ E-mail: D.B. Karp – [email protected]
1e studied the behavior of a particular case H q, p,q of Fox’s H function in the neighborhood of thispoint in our recent papers [6, 7] (as Braaksma’s parameter µ is called ∆ in our papers following [9],the case µ = 0 was given the denomination ”delta-neutral”). In particular, we found a convergentinverse factorial series for the integrand of the delta-neutral H q, p,q function and utilized it to derivea power series expansion of this function in a neighborhood of the singular point z = β − . We alsodeduced recursive and determinantal formulas for the coefficients of this expansion. However, ourwork contained an error caused by an incorrect citing and use of Nørlund’s results in [6, Theorem 2].The consequence of this error was omission of the condition that the scaling factors α i , β j in (1)should all be strictly greater than 1 / α i , β j . This is done insection 3. In order to obtain these results we needed a slight generalization of a particular case of[1, Theorem 2], which we establish in section 2. As some parts of Braaksma’s work do not seem tobe well-understood we also rewrite some of his ideas here. As we will need to modify some parts of Braaksma’s proof, for reader’s convenience we adhere tothe notation of [1]. Define H m,np,q (cid:18) z (cid:12)(cid:12)(cid:12)(cid:12) ( α , a ) , . . . ( α p , a p )( β , b ) , . . . ( β q , b q ) (cid:19) = H ( z ) = 12 πi Z C h ( s ) z s ds, (1)where z s = exp { s (log | z | + i arg z ) } , and arg z may take any real value. Here h ( s ) = Q nj =1 Γ(1 − a j + α j s ) Q mj =1 Γ( b j − β j s ) Q qj = m +1 Γ(1 − b j + β j s ) Q pj = n +1 Γ( a j − α j s )= Q pj =1 Γ(1 − a j + α j s ) Q qj =1 Γ(1 − b j + β j s ) | {z } h ( s ) × π m + n − p Q pj = n +1 sin[ π ( a j − α j s )] Q mj =1 sin[ π ( b j − β j s )] | {z } h ( s ) . The contour C is the right loop separating the poles s = ( b j + ν ) /β j , j = 1 , . . . , m , ν = 0 , , . . . (lying to the right of C , hence inside the contour) from the poles s = ( a j − − ν ) /α j , j = 1 , . . . , n , ν = 0 , , . . . (lying to the left of C , hence outside the contour). Denote µ = q X j =1 β j − p X j =1 α j , β = p Y j =1 α α j j q Y j =1 β − β j j , (2)and δ = m X j =1 β j − p X j = n +1 α j π. (3)Next, by [1, (4.26),(4.27),p.270] Q ( z ) = X residues of h ( s ) z s in the points s = ( a j − − ν ) /α j , j = 1 , . . . , n, ν = 0 , , . . . (4)2 ( z ) = X residues of h ( s ) z s in the points s = ( a j − − ν ) /α j , j = 1 , . . . , p, ν = 0 , , . . . (5)In [1, Lemma 4, p.263] Braaksma uses sin( z ) = e iz (1 − e − iz ) / (2 i ) to expand: h ( s ) = (2 πi ) m + n − p Q pj = n +1 e iπ ( a j − α j s ) (cid:0) − e − iπ ( a j − α j s ) (cid:1)Q mj =1 e iπ ( b j − β j s ) (cid:0) − e − iπ ( b j − β j s ) (cid:1) = (2 πi ) m + n − p e iπ { P pj = n +1 a j − P mj =1 b j } | {z } c e iπs { P mj =1 β j − P pj = n +1 α j } | {z } = e iγ s Q pj = n +1 (cid:0) − e iπ ( α j s − a j ) (cid:1)Q mj =1 (cid:0) − e iπ ( β j s − b j ) (cid:1) . For sufficiently large positive ℑ ( s ) we will have ℜ (2 iπ ( β j s − b j )) < | e iπ ( β j s − b j ) | <
1, andwe can expand (cid:0) − e iπ ( β j s − b j ) (cid:1) − in geometric series:11 − e iπ ( β j s − b j ) = ∞ X ν =0 e iπν ( β j s − b j ) = ∞ X ν =0 e − iπνb j | {z } B j,ν e iπνβ j s . Expanding we get: p Y j = n +1 (cid:16) − e iπ ( α j s − a j ) (cid:17) = 1 − A n +1 e iπα n +1 s − A n +2 e iπα n +2 s − . . . − A p e iπα p s + . . . + A n +1 A n +2 · · · A p e iπ ( α n +1 + α n +2 + ··· + α p ) s , where A k = e − iπa k . Multiplying all these expressions yields: h ( s ) = c e iγ s (cid:0) c e iγ s + ˆ c e iγ s + · · · (cid:1) with 0 < γ < γ < · · · , where γ = π nX mj =1 β j − X pj = n +1 α j o (6)and γ = 2 π min( α n +1 , α n +2 , . . . , α p , β , . . . , β m ) . (7)If ℑ ( s ) is negative and of large absolute value, we can use sin( z ) = − e − iz (1 − e iz ) / (2 i ) to similarlyexpand h ( s ) = d e − iγ s (cid:16) d e − iγ s + ˆ d e − iγ s + · · · (cid:17) , where d = ( − πi ) m + n − p e iπ { P mj =1 b j − P pj = n +1 a j } = (2 π ) m + n − p ) /c , and d j = 1 /c j . In [1, Definition I,p.265] Braaksma takes the (unordered) multi-set { . . . , − γ , − γ , − γ , γ , γ , γ , . . . } , sorts it in ascending order, removes the repeated elements and calls the new set { . . . , δ − , δ − , δ , δ , δ , . . . } with δ = γ . We next assume γ = δ = 0 . Then δ j = − δ − j and δ j > j = 1 , , . . . , and, according to [1,Definition II,p.265], we see that κ = 0 (by definition κ is the index such that δ κ = − δ ). Further,if an integer r > C r = c r , D r = 0; if r <
03e have case b) and C r = 0, D r = − d r ; if r = 0, then we have case c) and C = c , D = − d = − (2 π ) m + n − p ) /c . Note that the above formulas imply that (this is a slight modification of [1,(4.10),p.266]) h ( s ) = ∞ X j =0 c j e iδ j s , c j = c ˆ c j for j = 1 , , . . . (8)for ℑ ( s ) >
0, and h ( s ) = ∞ X j =0 d j e − iδ j s c j = d ˆ d j for j = 1 , , . . . (9)for ℑ ( s ) <
0; each series obviously uniformly converges: first for ℑ ( s ) ≥ ǫ >
0, while the second for ℑ ( s ) ≤ − ǫ <
0. Hence, c j , d j can be viewed as generalized Fourier coefficients in the upper andlower half-planes, while δ j are ”generalized frequencies”.Having made these preparations we can formulate (a part of) [1, Theorem 3,p.280] as follows. Theorem 1
Suppose µ = γ = 0 ( see (2) and (6) for definitions ) . Then H ( z ) can be continuedanalytically into the sector ∆ − = { z : − γ < arg( z ) < } , where γ is defined in (7) , by H ( z ) = 12 πi Z D h ( s ) { h ( s ) − c } z s ds − V ( z ); and into the sector ∆ + = { z : 0 < arg( z ) < γ } by H ( z ) = 12 πi Z D h ( s ) { h ( s ) − d } z s ds − V ( z ); the contour D starts at s = − i ∞ + σ and terminates at s = i ∞ + σ , σ is arbitrary real number,leaves the points s = ( a j − ν − /α j , j = 1 , . . . , p , ν = 0 , , . . . ( the poles of h ( s )) on the leftand the points s = ( b j + ν ) /β j , j = 1 , . . . , m , ν = 0 , , . . . , which do not coincide with the points ( a j − ν − /α j , j = 1 , . . . , p , ν = 0 , , . . . , ( the poles of h ( s ) which are not poles of h ( s )) on theright. The function V ( z ) is the sum of residues of h ( s ) z s in the points which are poles of both h ( s ) and h ( s ) . This function is equal to zero if b j + νβ j = a h − − λα h for all j = 1 , . . . , m , h = n + 1 , . . . , p , ν, λ = 0 , , , . . . Moreover, the function H ( z ) can be continued analytically from the sector ∆ − to the domain | z | > β − by the formula H ( z ) = Q ( z ) − c P ( z ) , and from the sector ∆ + to the domain | z | > β − by the formula H ( z ) = Q ( z ) − d P ( z ) . Here Q and P are defined in (4) and (5) , respectively. Both series (4) and (5) converge for | z | > β − . − and ∆ + are analytic in the ”large” sector ∆ − ∪ [0 , ∞ ) ∪ ∆ + . First, we have to require c = d . Writing the definitions of c and d and performing some simple calculations we arrive atthe condition: p X j = n +1 a j − m X j =1 b j = p − m − n η, where η is any integer. In this case we have c = d = (2 π ) m + n − p e iπ ( m + n − p ) / e iπ ( p − m − n ) / e iπη = ( − η (2 π ) m + n − p . In particular, if n = 0, m = q and η = 1 we obtain p X j =1 a j − q X j =1 b j = p − q , which corresponds to the case considered in [7]. Now in this particular situation ( µ = δ = 0, c = d ) we obtain a strengthened version of [1, Lemma 4a, p.265]: Lemma 1
Suppose µ = δ = 0 , c = d . Then there exists a positive constant K = K ( ǫ ) ,independent of s , such that for any ǫ > | ( h ( s ) − c ) e − iδ s | ≤ K if ℑ ( s ) ≥ ǫ and | ( h ( s ) − c ) e iδ s | ≤ K if ℑ ( s ) ≤ − ǫ. For the proof note that c = d and use (8) and (9) to get straightforward term-by-term estimates,similarly to the proof of [1, Lemma 4a, p.265]. Note, however, that we get a larger difference 2 δ between the exponents of the multipliers e − iδ s and e iδ s in the first and second bounds, than thatin [1, Lemma 4a, p.265]. This becomes possible by virtue of the condition c = d .We now formulate an extension of a particular case of [1, Theorem 2, p.280] for the values ofparameters µ = δ = 0 and c = d = ( − η (2 π ) m + n − p ( η ∈ Z is arbitrary). Theorem 2
Suppose µ = δ = 0 , where µ and δ are defined in (2) and (3) , respectively, and c = d . Then c = d = ( − η (2 π ) m + n − p , where η = p X j = n +1 a j − m X j =1 b j − p − m − n ∈ Z . Furthermore, the function H m,np,q ( z ) defined in (1) can be continued analytically to the sector ∆ γ = { z : − γ < arg( z ) < γ } , (10) where γ is defined in (7) by means of H ( z ) = 12 πi Z D h ( s ) { h ( s ) − c } z s ds − V ( z ); (11) the contour D and the function V ( z ) have been defined in Theorem 1. Moreover, the function H ( z ) can be continued analytically from this sector to the domain | z | > β − on the Riemann surface ofthe logarithm by the formula H ( z ) = Q ( z ) − c P ( z ) , Here Q and P are defined in (4) and (5) , respectively. Both series (4) and (5) converge for | z | > β − . roof. We follow the proof of [1, Theorem 2, p.280] for r = 1 and [1, (6.10), p.280] taking theform: f ( s ) = h ( s ) β − s { h ( s ) − c } e − iγ s , so that, in view of the first bound in Lemma 1, we can apply [1, Lemma 6, p.271] with βze iγ playing the role of z . Then [1, Lemma 6, p.271] applies for arg( z ) > − γ . Instead of formula [1,(6.11), p.280] we take f ( s ) = h ( s ) β − s { h ( s ) − c } e iγ s . In view of the second bound in Lemma 1, we can now apply [1, Lemma 6a, p.277] with βze − iγ playing the role of z and hence [1, Lemma 6a, p.277] applies for arg( z ) < γ . This leads tothe validity of [1, (6.12), p.280] for − γ < arg( z ) < γ . The remaining part of the proof of [1,Theorem 2, p.280] remains intact. (cid:3) For the function H q, p,q ( z ) ( m = q , n = 0) we have Q ( z ) = 0 by (4), so that if µ = 0 (and hencein this case δ = πµ = 0) and η = p X j =1 a j − q X j =1 b j + q − p ∈ Z , (12)(note that η = α − /
2, where α is defined in [1, (3.24)]), then H q, p,q ( z ) is analytic in the sector − γ < arg( z ) < γ , where γ is defined in (7), its analytic continuation to this sector is given by (11) (with c =( − η (2 π ) q − p ) and from this sector to the domain | z | > β − by( − η +1 (2 π ) q − p X residues of h ( s ) z s in the points s = ( a j − − ν ) /α j , j ∈ { , . . . , p } , ν ∈ N , according to (5). We now focus on the function H q, p,q ( z ) with µ = 0. For further convenience we also apply the changeof variable s → − s in the integral (1) to obtain for | z | < β − the definition H q, p,q (cid:18) z (cid:12)(cid:12)(cid:12)(cid:12) ( α , a ) , . . . ( α p , a p )( β , b ) , . . . ( β q , b q ) (cid:19) = 12 πi Z C ′ Q qj =1 Γ( b j + β j s ) Q pj =1 Γ( a j + α j s ) z − s ds, (13)where the contour C ′ = − C , it is the left loop starting at −∞ − ik , terminating at −∞ + ik and encompassing all the poles s = ( b j − ν ) /β j , j = 1 , . . . , q , ν = 0 , , . . . (so that k > ℑ ( b j /β j ), j = 1 , . . . , q ). To be consistent with [6, 7] denote ρ = β − > . Assuming that η = 1, we see by Theorem 2 that the function H q, p,q ( ρt ) is analytic in the domain G := U − ∪ ∆ γ , where U − = { t : | t | < } \ ( − ,
0] denotes the unit disk cut along the interval( − ,
0] (we need a cut in view of the branch point at t = 0) and∆ γ = {− γ < arg( t ) < γ } , γ = 2 π min( α , . . . , α p , β , . . . , β q ) ,
6s the sector defined in (10) (note that change of variable z → ρt does not alter this sector). Hence,for arbitrary complex σ we have the power series expansion φ ( t ) = t − σ H q, p,q ( ρt ) = ∞ X n =0 V n ( σ )(1 − t ) n (14)convergent in the disk | − t | < R , where R is the distance from t = 1 to the boundary of thedomain G . Elementary geometry shows that R = 1 if γ ≥ π/ α , . . . , α p , β , . . . , β q ) ≥ /
6) and R = 2 sin( γ /
2) if γ < π/
3. Assume for a moment that γ ≥ π/
3. Multiplying the above expansion by t z − and integrating term by term (to be justifiedbelow) we get Z t z + σ − t − σ H q, p,q ( ρt ) dt = ∞ X n =0 V n ( σ ) Z t z + σ − (1 − t ) n dt = ∞ X n =0 V n ( σ ) n !( z + σ ) n +1 . On the other hand, by [8, Theorem 6] Z t z − H q, p,q ( ρt ) dt = ρ − z Z ρ u z − H q, p,q ( u ) du = ρ − z Q qj =1 Γ( β j z + b j ) Q pj =1 Γ( α j z + a j )or W ( z ) = ρ − z Q qj =1 Γ( β j z + b j ) Q pj =1 Γ( α j z + a j ) = ∞ X n =0 V n ( σ ) n !( z + σ ) n +1 . (15)This inverse factorial series was derived in [6, Theorem 1], where the coefficients were found explic-itly. However, the condition γ ≥ π/ η = 1. Our proof will be based on a theorem due to Nørlund. To formulate this theorem weneed to define the Hadamard order of a function f = P n ≥ f n x n holomorphic inside the unit disk | x | <
1. According to [12, 23(1),p.46] the Hadamard order of f on the circle | x | = 1 is defined by: ω = lim sup n →∞ log | nf n | log( n ) . (16)Nørlund proved the following theorem (the formulation below is a combination of [12, Theorem § § Theorem 3
Suppose φ ( t ) = ∞ X s =0 a s (1 − t ) s is holomorphic for | − t | < and has a finite order on the circle | − t | = 1 . Then F ( w ) = Z t w − φ ( t ) dt can be expanded in the inverse factorial series F ( w ) = ∞ X s =0 a s s !( w ) s +1 convergent in a half-plane ℜ ( w ) > λ ( save the points w = 0 , − , − , . . . ) and divergent for ℜ ( w ) < λ .If the series P ∞ s =0 a s diverges, the convergence abscissa λ = h − , where h is the order of t − φ ( t ) on the the circle | − t | = 1 ; if P ∞ s =0 a s converges, then λ = h ′ − , where h ′ is the order of [ φ ( t ) − φ (0+)] /t on the circle | − t | = 1 . γ > π/
3. Then, according to the above, for some ε > < φ < π/ H q, p,q ( ρt ) is holomorphic in the domain∆( φ, ε ) = { t : | − t | ≤ ε and − π + φ ≤ arg( t ) ≤ π − φ } , which is larger than the disk | − t | <
1. We want to apply Theorem 3 with φ ( t ) = t − σ H q, p,q ( ρt )and w = z + σ . Hence we need to determine the order of φ ( t ) t − = t − σ − H q, p,q ( ρt ) on the circle | − t | = 1 and the order of [ φ ( t ) − φ (0+)] /t when φ (0+) = ∞ . First recall the shifting property[9, (2.1.5)] t − σ H q, p,q (cid:18) ρt (cid:12)(cid:12)(cid:12)(cid:12) ( α , a ) , . . . ( α p , a p )( β , b ) , . . . ( β q , b q ) (cid:19) = ρ σ H q, p,q (cid:18) ρt (cid:12)(cid:12)(cid:12)(cid:12) ( α , a − σα ) , . . . ( α p , a p − σα p )( β , b − σβ ) , . . . ( β q , b q − σβ q ) (cid:19) Next, suppose P σ = { ( s, r ) : s = pole of h σ ( s ); r = its multiplicity } , where s runs over all distinct poles of the integrand of t − σ H q, p,q ( ρt ) which is given by h σ ( s ) = Q qj =1 Γ( b j − σβ j + β j s ) Q pj =1 Γ( a j − σα j + α j s ) , and r stands for the multiplicity of the corresponding pole. We need the following definitions:denote by P ′ σ the set P σ where the elements of the form ( m, m ∈ Z ≤ , have been removed (inother words simple poles at non-positive integers are removed); and defineˆ β ( σ ) = max {ℜ ( s ) : ( s, r ) ∈ P ′ σ } . (17)Note that ˆ β ( σ ) = max ≤ j ≤ q ℜ ( σ − b j /β j ) except when this maximum is attained for σ − b j /β j ∈ Z ≤ and the corresponding pole is simpe. It is also convenient to write P σ, for the set of the firstcomponents of the elements of P σ (i.e. the set of poles of h σ ( s ) without multiplicities). Similarlyfor P ′ σ . Lemma 2
Let φ ( t ) be defined in (14) .If the series P n ≥ V n ( σ ) diverges, then the Hadamard order h of t − φ ( t ) on the circle | − t | = 1 is given by h = ˆ β ( σ ) + 1 .If the series P n ≥ V n ( σ ) converges, then the Hadamard order h ′ of [ φ ( t ) − φ (0+)] /t on the circle | − t | = 1 is given by the same expression h ′ = ˆ β ( σ ) + 1 . Proof.
Denote I σ := −P σ, \ P ′ σ, . By definition I σ ⊂ Z ≥ . The residue theorem applied to the definition of t − σ H q, p,q ( ρt ) gives as t → φ ( t ) = t − σ H q, p,q ( ρt ) = ρ σ X res s ∈P ′ σ, h σ ( s )( ρt ) − s + ρ σ X res s ∈P σ, \P ′ σ, h σ ( s )( ρt ) − s = (log t ) r − l X k =1 C k t − ˆ b k (1 + o (1)) + X j ∈ I σ ⊂ Z ≥ A j t j , (18)where ˆ b k = σ − b j k /β j k , k = 1 , , . . . , l , are the elements of P ′ σ, with ℜ (ˆ b k ) = ˆ β ( σ ) and havingmaximal multiplicity (which we denoted by r ) among all poles with real part ˆ β ( σ ). The precise8alues of the non-zero constants C , . . . , C l can be found in [9, Theorem 1.5], but they are immaterialfor our purposes here.To determine the order note first that if the series P n ≥ V n ( σ ) converges, then lim t → φ ( t ) < ∞ by Abel’s theorem. Hence, by (18) this series diverges if ˆ β ( σ ) > β ( σ ) = 0 and r > β ( σ ) = 0, r = 1 and ℑ (ˆ b k ) = 0 for some k ∈ { , . . . , l } (because no finite limit lim t → φ ( t )exists in all these cases). As ˆ b k ∈ P ′ σ, the case ˆ b k = 0 is impossible by definition of P ′ σ . Hence, weconclude that P n ≥ V n ( σ ) diverges if ˆ β ( σ ) ≥
0. To determine the of Hadamard order of t − φ ( t ) inthese situations we apply [3, Theorem VI.4] with f ( x ) = (1 − x ) − σ − H q, p,q ( ρ (1 − x ))and ζ = 1. This gives the following asymptotic relation for the power series coefficients V n ( σ + 1) V n ( σ + 1) = A + l X k =1 D k n ˆ b k (log n ) r − (1 + o (1)) + o (1) (19)with some complex constants D k and o (1) being different in each term. By definition of theHadamard order (16) the above asymptotic formula leads immediately to h = ˆ β ( σ ) + 1.Next, we assume that the series P n ≥ V n ( σ ) converges, so that lim t → φ ( t ) < ∞ . By the aboveargument this is only possible if ˆ β ( σ ) <
0. Then, by (18) we havelim t → φ ( t ) = A ( A may vanish). The last term in (18) for t − ( φ ( t ) − A ) takes the form X j ∈ I σ \{ }⊂ Z > A j t j − which is holomorphic around t = 0 and does not affect the order of t − ( φ ( t ) − A ) (see [12,section 26]). Hence (19) with A removed holds true again, and the order h ′ of t − ( φ ( t ) − A ) stillequals h ′ = ˆ β ( σ ) + 1. (cid:3) The coefficients V n ( σ ) have been computed in [6, Corollary 1] and [7, Theorem 1] via rearrange-ment of Poincar´e asymptotics of W ( z ) defined in (15) as z → ∞ into inverse factorial series (thismethod dates back to Stirling). Putting these facts together we arrive at the following statements. Theorem 4
Suppose γ = 2 π min( α , . . . , α p , β , . . . , β q ) > π/ , µ = 0 and η = 1 , where µ and η are defined in (2) and (12) , respectively. Then the function t − σ H q, p,q (cid:18) ρt (cid:12)(cid:12)(cid:12)(cid:12) ( α , a ) , . . . ( α p , a p )( β , b ) , . . . ( β q , b q ) (cid:19) defined in (13) , analytic in the sector − γ < arg( z ) < γ by Theorem 2, can be developed inconvergent power series (14) with coefficients given by V n ( σ ) = (2 π ) ( q − p ) / Y qk =1 β b k − / k Y pj =1 α / − a j k n X k =0 ( − k l n − k k !( n − k )! B ( n +1) k (1 − σ ) , (20) where ρ = β − and β is defined in (2) , B ( n +1) k ( · ) is Bernoulli-Nørlund polynomial [13, (1)]. Thecoefficients l r satisfy the recurrence relation ( with l = 1) l r = 1 r r X m =1 q m l r − m , q m = ( − m +1 m + 1 q X k =1 B (1) m +1 ( b k ) β mk − p X j =1 B (1) m +1 ( a j ) α mj . (21)9he following theorem is a corrected and refined version of [6, Theorem 5]. Theorem 5