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As the sequence {(z)n+1}n=0 is an asymptotic sequence for |z|→∞ in {z : | arg(z)| < π − ε}, the series (1) is an asymptotic series as |z|→∞ −k regardless of its convergence. The idea to convert the Poincar´e asymptotic series akz into a convergent inverse factorial series goes back at least to 1912 paper of G.N. Watson [44]. It has been revived more recently in [10] and [46] and combined with Borel-LaplaceP summation in the former reference. Inverse factorial series play an important role in solution of diﬀerence equations [9, 29]. In this note we exploit similar ideas (resummation of Poincar´etype series into convergent inverse factorial series) to derive the inverse factorial series expansion of the function p Γ(A z + a ) W (z)= ρ−z k=1 k k (2) q Γ(B z + b ) Q j=1 j j with explicit formulas for the coeﬃcients andQ precise determination of the convergence do- p Ak q −Bj main. Here Ak, Bj are positive, while ak, bj are complex numbers, ρ = k=1 Ak j=1 Bj . This expansion was instrumental in the study of the behavior of Fox’s H function Hp,0(t) (de- Q Qq,p ﬁned below) in the neighborhood of the singular point t = ρ undertaken by us in [18]. Hence, this note also ﬁlls a gap in the proof of [18, Theorem 1]. The problem of expanding the func- tion W (z) and its particular cases in inverse factorial series has been considered previously by a number of authors. Probably, the ﬁrst appearance of such expansion is in Ford’s book [13], where the inverse factorial series for W (z) with p = q = 2, A1 = A2 = B1 = B2 =1/2 and a1 + a2 = b1 + b2 was found and proved to be asymptotic. No explicit formulas for the coeﬃcients were given. This was improved by Van Engen in [42], where the author found the coeﬃcients in Ford’s expansion and removed the restriction a1 + a2 = b1 + b2. The general ratio W (z) was ﬁrst considered by Wright in the sequel [47, 48]. He proved that there exists a series in reciprocal gamma functions asymptotic to the function W (z) under very general assumptions. Wright only gave a formula for the ﬁrst coeﬃcient, but mentioned that fur- ther coeﬃcients could also be computed. Similar result was later proved by Hughes in [15] p q under the natural additional restriction k=1 Ak = j=1 Bj and using the standard inverse factorial series (1). In his milestone work [30] Nørlund deduced an inverse factorial series P P expansion of the function (2) when p = q and Ak = Bk = 1, k =1,...,p, and proved its con- vergence in the intersection of the half planes ℜ(z + ak) > 0, k =1,...,p. He also gave two diﬀerent methods to compute the coeﬃcients in this expansion. In a series of papers [35]-[37] Riney studied the function (2) for p ≤ q + 1 and Aj = Bk = 1, j = 1,...,p, k = 1,...,q. He gave an asymptotic series for this function in terms of gamma ratios, of which standard factorial series is a particular case, and presented several methods to compute the coeﬃ- cients. Riney’s investigations were complemented by van der Corput [41], who considered the opposite case q ≤ p + 1, and Wright [49], who suggested further methods for calculating the coeﬃcients. Braaksma [3] again considered the general case of (2) and proved that there exists a series in reciprocal gamma functions asymptotic to W (z). He also gave an explicit formula for the principal term. This result of Braaksma (which is just a technical tool in his deep investigation of Mellin-Barnes integrals) is, in fact, a modiﬁcation of the earlier work by Wright [47, 48] mentioned above. A survey of some of the above work is given in section 2.2 of the book [33] by Paris and Kaminski, where one can also ﬁnd explicit proofs and several examples. Independently, Gupta and Tang [14] presented a series in gamma ratios for W (z) p q when k=1 Ak = j=1 Bj and gave certain recursions for computing the coeﬃcients. They P P 2 also claim convergence but gave no real proof of this claim. Further details about their work can be found in the introduction to our paper [18]. In the present paper we combine some ideas from [14] with Borel-Laplace summation to give a rigorous proof of convergence and formulas for the coeﬃcients for the inverse p q factorial series expansion of W (z) under the assumption k=1 Ak = j=1 Bj. This is done in section 3 of this paper. Furthermore, in section 4 we apply the inverse factorial series for a simplest particular case of (1) to derive a presumablyP new identPity for the Nørlund- Bernoulli polynomials. The main results presented in sections 3 and 4 are preceded by the preliminaries expounded in section 2.

2. Preliminaries. A function f(z) of a complex variable z is said to possess a Poincar´e type asymptotic expansion in an unbounded domain D if

n−1 α f(z)= k + R (z) (3) zk n Xk=0 −n and Rn(z)=Ø(z ) as D ∋ z →∞. If this holds for all natural n it is customary to write

∞ α f(z) ∼ k as z →∞ in D. zk Xk=0 The asymptotic expansion of f(z) is said to be Gevrey-1 (or Gevrey of order 1), if there exist the numbers M,τ > 0 such that for all z ∈ D and all positive integers n the error term Rn(z) in (3) satisﬁes [23, Deﬁnition 5.21], [8, Deﬁnition 4.130] Mτ nn! |R (z)| ≤ . (4) n |z|n

It is convenient to introduce the following class of functions. Deﬁnition. We will say that f belongs to the class G if f possesses a Gevrey-1 expansion in some right half-plane ℜz>λ = λ(f). Let us list some properties of the class G required in the sequel. The proofs are either straightforward from the above deﬁnition or are given reference to. Property 1 (linearity). If f ∈ G and g ∈ G, then f +g ∈ G and af(z) ∈ G for arbitrary complex a =6 0. Property 2 (shifting and dilating invariance). If f ∈ G, then f(Az + a) ∈ G for A> 0 and arbitrary complex a. Property 3 (invariance under taking exponential) If f(z) ∈ G, then exp{f(z)} ∈ G (see [43, pp. 288, 293] for a proof). Property 4. If f ∈ G and α0 = 0 in (3), then zf(z) ∈ G. Property 5. If f ∈ G, then f(z)/z ∈ G. Property 6. If ϕ(z) is holomorphic in the neighborhood of z = 0, then f(z)= ϕ(a/z) ∈ G for arbitrary complex a =6 0. To demonstrate the last property apply Cauchy estimates to the Taylor coeﬃcients ak of k ϕ(z) to get |ak| ≤ M/r , where r is strictly less than the radius of convergence of the Taylor

3 series of ϕ(z) at z = 0 and M = max |ϕ(w)|. This leads to the following estimate of the |w|≤r Taylor remainder

∞ n ∞ n n k M|z| j M|z| M|z| |Qn(z)| = akz ≤ |z/r| = ≤ rn rn(1 −|z/r|) rn(1 − β) k=n j=0 X X for |z| ≤ βr and arbitrary β ∈ (0, 1). Hence, n−1 M|a/r|n |a| ϕ(a/z)= a (a/z)k + Q (a/z) and |Q (a/z)| ≤ for |z| ≥ , k n n |z|n(1 − β) βr Xk=0 so that (4) is trivially satisﬁed. We will need the next well-known lemma relating the coeﬃcients of an asymptotic series of a function with those of its exponential. Essentially, the result contained in this lemma appeared in [24, Appendix]. Later, an independent derivation was given in [16, Lemma 1]. It has also been discussed recently in [34], where further references are given. Surprisingly, references [24] and [16] do not appear in [34].

∞ −k ∞ −r Lemma 1 Suppose g(z) ∼ k=1 ukz as z → ∞. Then exp{g(z)} ∼ r=0 vrz , where the coeﬃcients are found from P P 1 r v =1, v = ku v . 0 r r k r−k Xk=1 Alternatively,

k1 k2 kr r n u1 u2 ··· ur 1 vr = = uki . k1!k2! ··· kr! n! k1+2k2+···+rkr=r n=1 k1+k2+···+kn=r i=1 kXi≥0 X kXi≥1 Y

Remark. Nair [24, section 8] found a determinantal expression for vr which in our notation takes the form u (i − j + 1)(i − 1)!/(j − 1)!, i ≥ j, det(Ω ) i−j+1 v = r , Ω = [ω ]r , ω = −1, i = j − 1, r r! r i,j i,j=1 i,j 0, i

Theorem 1 (Watson-Nevanlinna-Sokal) Suppose f(z) is holomorphic in Cr = {z : ℜ(1/z) >r−1} and can be written as

n−1 k f(z)= αkz + Rn(z) (5) Xk=0 n n with the error term satisfying |Rn(z)| ≤ Mτ n!|z| , where M is independent of n and z ∈ Cr. Then its Borel transform ∞ α tk B(t)= k k! Xk=0 4 + converges for |t| < 1/τ and can be extended analytically to the domain Sτ = {t : dist(t, R ) < 1/τ} to a function satisfying |B(t)|

∞ 1 f(z)= e−t/zB(t)dt. (7) z Z0

′ Conversely, if B(t) is holomorphic in Sτ ′ (τ < τ) and satisﬁes (6), then the function f(z) deﬁned by the integral (7) is holomorphic in Cr and has Gevrey-1 a symptotic approximation (k) (5) with uniform error bound in Cr, where αk = B (t) |t=0.

The next theorem can be found in [28, Theorem VIII], [29, p. 267] and [45, Theorem 46.2].

Theorem 2 (Nørlund) Suppose Ω(z) satisﬁes the following conditions: 1) Ω(z) = a/z + v(z)/[z(z + 1)], where v(z) is holomorphic and bounded in some right half-plane ℜz > κ. ∞ −tz R+ 2) Ω(z)= 0 e B(t)dt, where B(t) is holomorphic in the domain Sη = {t : dist(t, ) < −kt η} for some η > 0 and satisﬁes in Sη the condition lim e B(t)=0 for some positive k. R t→∞ Then Ω(z) can be expanded in the inverse factorial series

∞ b Ω(z)= s z(z + 1) ··· (z + s) s=0 X convergent in some right half-plane ℜz>λ excluding the points z =0, −1, −2,....

To determine the abscissa of convergence λ we need the following notion due to Hadamard [28, pp.333-334]: a function f(z)= a0 + a1z + ··· holomorphic in the unit disk |z| < 1 has the order κ on the circle |z| = 1 if

log |na | κ = lim sup n . n→∞ log(n)

The next theorem [28, Theorem III] relates the order of the so-called generating function ϕ of an inverse factorial series Ω to its abscissa of convergence.

Theorem 3 (Nørlund) Suppose the next representation holds for suﬃciently large val- ues of ℜz: ∞ b 1 Ω(z)= s = tz−1ϕ(t)dt, (8) (z) s=0 s+1 0 X Z and assume that the order of t−1ϕ(t) on the circle |1 − t| = 1 is equal to κ. If κ > 1, then the abscissa of convergence λ of the inverse factorial series in (8) is equal to κ −1, otherwise λ ≤ κ − 1.

5 Further, Nørlund showed in [28, (7), page 339] that a function f(t) holomorphic in |1 − t| ≤ 1 except for a singularity at t = 0 and representable in the form

m ai r f(t)= t (ψi,0(t)+ ψi,1(t) log(t)+ ··· + ψi,r(t) log (t)) (9) i=1 X in the neighborhood of t = 0 with ℜ(a1) ≤ℜ(a2) ≤···≤ℜ(am) has the order κ = −ℜ(a1). It is assumed that ψi,j(t), i =1,...,m, j =0,...,r, are holomorphic around t = 0 and such that for each i =1,...,m at least one of the numbers {ψi,0(0),...,ψi,r(0)} is diﬀerent from zero. Integer nonnegative ai such that r = 0 (no logarithmic terms) must be excluded from the determination of order. Our main tool is the following theorem.

Theorem 4 Let f(z) be holomorphic in some right half-plane ℜz > κ and suppose that log(zf(z)) ∈ G. Then for any complex β the function f(z) can be expanded in the inverse factorial series ∞ d f(z)= s (z + β) s=0 s+1 X convergent in some right half-plane ℜz>λ.

Proof. By Properties 3, 2 and 5, respectively, ζf(ζ) ∈ G, (ζ − β)f(ζ − β) ∈ G and f(ζ) ∈ G. Then ζf(ζ − β)=(ζ − β)f(ζ − β)+ βf(ζ − β) ∈ G by Properties 2 and 1. Therefore, n−1 β ζf(ζ − β)= k + R (ζ) (10) ζk n Xk=0 with the remainder Rn(ζ) bounded according to (4). Now put f1(ζ)= f(ζ − β) and rewrite (10) as f (1/w) n−1 1 = β wk + R (1/w) w k n Xk=0 with the error bound of the form

n n |Rn(1/w)| ≤ Mτ n!|w| .

According to Theorem 1 ∞ f (1/w) 1 1 = e−t/wB(t)dt w w Z0 or ∞ −tζ f1(ζ)= e B(t)dt Z0 and B(t) satisﬁes condition 2 of Theorem 2. As (ζ − β)f1(ζ) ∈ G, we have by deﬁnition of G: γ A (ζ − β)f (ζ)= γ + 1 + T (ζ), |T (ζ)| ≤ . 1 0 ζ 2 2 |ζ|2

6 This implies that γ v(ζ) f (ζ)= 0 + , 1 ζ ζ(ζ + 1) where (γ β + γ )(1 + β) ζ(ζ + 1)T (ζ) v(ζ)= γ β + γ + 0 1 + 2 . 0 1 ζ − β ζ − β

The above estimate for T2(ζ) immediately leads to conclusion that v(ζ) is bounded outside of some neighborhood of ζ = β. Thus, the ﬁrst condition of Theorem 2 is also satisﬁed and

∞ d f (ζ)= s . 1 (ζ) s=0 s+1 X Substituting back z = ζ + β yields

∞ d f(z)= f (z + β)= s . 1 (z + β) s=0 s+1 X The claim regarding convergence follows by Theorem 2. We will write Bj(x) for the j-th Bernoulli polynomial [32, 24.2.3], deﬁned by the gener- ating function text ∞ tn = B (x) , |t| < 2π. et − 1 n n! n=0 X Further, Bj = Bj(0) are Bernoulli numbers [32, 24.2.1].

Lemma 2 The function Pa(z) = log Γ(z + a) − (z + a − 1/2) log z + z belongs to the class G and 1 ∞ (−1)jB (a) P (z) ∼ log(2π)+ j (11) a 2 j(j − 1)zj−1 j=2 X as |z|→∞ in the domain | arg z| < π − δ, 0 <δ<π.

Proof. Formula (11) is Hermite’s asymptotic expansion for log Γ(z + a) [25, (1.8)]. We only need to proof its Gevrey-1 character. According to [33, (2.1.1) and (2.1.5)] and [12, (10),(14)], the Binet function 1 1 P (z) − log(2π) = log Γ(z) − (z − 1/2) log z + z − log(2π) 0 2 2 satisﬁes the relation

1 n−1 B P (z) − log(2π)= 2r + R (z), 0 2 2r(2r − 1)z2r−1 2n−1 r=1 X in the domain | arg z| < π and for z = |z|eiθ with |θ| < π the next inequality holds:

|B || sec2n(θ/2)| |R (z)| ≤ 2n . 2n−1 2n(2n − 1)|z|2n−1

7 In view of the asymptotic equality [11, Corollary 1], [12, (16)] 2(2n)! |B | = (1 + o(1)), n →∞, 2n (2π)2n we conclude that P0(z) ∈ G. It is straightforward to check that a 1 a P (z)= P (z + a) − a + z log 1+ + a − log 1+ . a 0 z 2 z a a Note that P0(z + a) ∈ G, log(1 + z ) ∈ G, z log(1 + z ) ∈ G according to the Properties 2, 6 and 4, respectively. It is left to apply Property 1 to conclude that Pa(z) ∈ G. Uniqueness of the asymptotic expansion completes the proof. The next lemma contains a corrected version of a formula contained in [14] which, in a diﬀerent form, was already presented in [2] (see also [1, 8.5.1]). p q Lemma 3 Let Ai, Bj > 0 satisfy i=1 Ai = j=1 Bj and ai, bj, i = 1,...,p, j = 1,...,q, be arbitrary complex numbers. Then for each 0 <δ<π the next asymptotic relation holds in the domain | arg z| < πP− δ, P zµ p Γ(A z + a ) ∞ C k=1 k k ∼ r , νρz q Γ(B z + b ) zr j=1 j j r=0 Q X where Q p q (p−q)/2 ak−1/2 1/2−bj ν = (2π) A Bj , (12) k=1 k j=1 p q q p Y Y p − q ρ = AAk B−Bj , µ = µ(a, ) = b − a + . (13) k j j k 2 j=1 ¯ j=1 kY=1 Y X Xk=1 The coeﬃcients Cr = Cr(A, B; a, ) are found from the recurrence ¯ 1 r C =1, C = Q (A, B; a, )C (14) 0 r r m r−m m=1 ¯ X or by other expressions contained in Lemma 1. Here p q (−1)m+1 B (a ) B (b ) Q (A, B; a, ) = m+1 k − m+1 j , (15) m m +1 Am Bm ¯ " k j=1 j # Xk=1 X and Bm(·) denotes the m-th Bernoulli polynomial. Furthermore, zµ p Γ(A z + a ) log k=1 k k ∈ G. νρz q Γ(B z + b ) Q j=1 j j ! Proof. From Hermite’s asymptotic formulaQ (11) we obtain by straightforward calculations: 1 m (−1)jB (a) log Γ(Az + a)=(Az + a − 1/2) log Az − Az + log(2π)+ j + R (z) 2 j(j − 1)(Az)j−1 m j=2 X 1 = Az log z +(A log A − A)z +(a − 1/2) log z +(a − 1/2) log A + log(2π) 2 m (−1)jB (a) + j + R (z), j(j − 1)Aj−1zj−1 m j=2 X

8 −m where Rm(z) = Ø(z ) in the sector | arg(z)| < π − δ for any δ > 0. For A > 0 Lemma 2 implies that log Γ(Az + a) − (Az + a − 1/2) log Az + Az ∈ G. Then employing the condition Ak = Bj we obtain

P P p ∞ Γ(Akz + ak) Qt(A, B; a, ) log k=1 ∼ z log ρ − µ log z + log ν + ¯, q Γ(B z + b ) tzt ( j=1 j j ) t=1 Q X Q where Qt(A, B; a, ) is given in (15). According to Property 1 ¯ zµ p Γ(A z + a ) log k=1 k k ∈ G. νρz q Γ(B z + b ) ( Q j=1 j j ) Q Formula (14) and other methods to compute Cr follow from Lemma 1.

3. Main results. To formulate our mains theorem we will need the non-central Stirling numbers of the ﬁrst kind [7, 8.5] deﬁned by

n n n n s (n, r)= (−1)k+r (σ) s(k,r)= (−1)k+r (σ) s(k,r), (16) σ k n−k k n−k Xk=r Xk=0 where s(n, j) denotes the ordinary Stirling number of the ﬁrst kind generated by

n x(x − 1) ··· (x − n +1)= s(n, j)xj. j=0 X Their ”horizontal” exponential generating function is given by [46, (A.2)]:

n n n n xls (n, l)= xl (−1)k+l (σ) s(k,l) σ k n−k Xl=0 Xl=0 Xk=0 n n k n n = (−1)k (σ) s(k,l)(−x)l = (σ) (x) =(σ + x) , k n−k k n−k k n Xk=0 Xl=0 Xk=0 k k l where the expansion (x)k =(−1) l=0 s(k,l)(−x) has been applied. Sometimes it is more convenient to use the ”vertical” generating function P ∞ s (n, l) ∞ s (n, l) ∞ n (σ) s(k,l) σ xn = σ xn = xn (−1)k+l n−k n! n! k!(n − k)! n=0 X Xn=l Xn=l Xk=l ∞ xk ∞ (σ) =(−1)l (−1)ks(k,l) n−k xn−k k! (n − k)! Xk=l Xn=k ∞ (−x)k ∞ (σ) 1 1 1 l =(−1)l s(k,l) m xm = log . k! m! l! (1 − x)σ 1 − x m=0 Xk=l X The non-central Stirling numbers were studied by Carlitz in [5, 6] using the symbol R1(n, l, σ)= sσ(n, l) and some years later also by Koutras [20]. Broder [4] considered them

9 for integer σ from the combinatorial viewpoint. Various formulas for these numbers are given in [7, 8.5]. Among other things, Carlitz found the double generating function [5, (5.4)] ∞ xn s (n, l)yl = (1 − x)−σ−y. σ n! l,nX=0 Using a formula from [22, section 6.43,p.134] this generating function leads to the connection formula [6, (7.6)] −l − 1 (−1)n−l(l + 1) s (n, l)= B(n+1)(1 − σ)= n−l B(n+1)(1 − σ), σ n − l n−l (n − l)! n−l γ where Bk (x) is the k-th Nørlund-Bernoulli polynomial (also known as the generalized Bernoulli polynomials) deﬁned by the generating function [31, (1)] tγext ∞ tk = B(γ)(x) . (17) (et − 1)γ k k! Xk=0 The main result of this note is the following theorem. p q Theorem 5 Let Ak, Bj > 0 satisfy k=1 Ak = j=1 Bj and σ, ak, bj, k = 1,...,p, j = 1,...,q, be complex numbers such that µ = 1, where µ is deﬁned in (13). Suppose further that ν and ρ are given by (12) andP (13), respectively.P Then the inverse factorial series expansion p Γ(A z + a ) ∞ ν n W (z)= ρ−z k=1 k k = C s (n, r) (18) q Γ(B z + b ) (z + σ) r σ j=1 j j n=0 n+1 r=0 Q X X converges for ℜz>α exceptQ at the points z = −σ − l, l ∈ N0. Here α denotes the real part of the rightmost pole of W (z) (obviously, α = − min{ℜ(ak/Ak): k = 1,...,p} if no cancelations of the numerator and denominator poles take place). Moreover, if ℜ(σ+α) > 0, then α is equal to the abscissa of convergence of the series (18). Here Cr = Cr(A, B; a, ) is ¯ deﬁned in (14), (15) and sσ(n, r) is the non-central Stirling number of the ﬁrst kind. Remark. The convergence domain in the above theorem can be described as follows: if the abscissa of the rightmost pole of W (z) is greater than the abscissa of the rightmost pole of the series on the right hand side, then the convergence abscissa is equal precisely to the abscissa of the rightmost pole of W (z); if the abscissa of the rightmost pole of W (z) does not exceed the abscissa of the rightmost pole of the series on the right hand side, then the convergence abscissa does not exceed the abscissa of the rightmost pole of W (z); if the rightmost poles on both sides are simple and coincide (i.e. σ = ak∗ /Ak∗ , where ∗ k = argmin{ℜ(ak/Ak): k =1,...,p}), then these poles should be ignored when calculating the convergence abscissa. Proof. As µ = 1 according to Lemma 3 log(zW (z)) ∈ G. Then, by Theorem 4 we −1 conclude that W (z) can be expanded in a series of inverse factorials [(z + σ)n+1] . As this series is also asymptotic for W (z) as z →∞, its coeﬃcients can be obtained by rearranging the Poincar´easymptotic expansion of W (z). According to Lemma 3 the latter is given by ∞ Cr(A, B; a, ) W (z) ∼ ν ¯, zr+1 r=0 X 10 where the coeﬃcients Cr(A, B; a, ) are deﬁned in (14), (15). Following [44] and [46, (4.1)] ¯ ∞ this asymptotic series can be re-expanded using the asymptotic sequence {1/(z)n+1}n=0 by applying [27, §30(6)] 1 ∞ (−1)ks(r + k,r) r+1 = , z (z)r+k+1 Xk=0 where s(n, j) is the Stirling number of the ﬁrst kind. Substituting and changing the order of summations we get ∞ (−1)n n W (z) ∼ ν (−1)rs(n, r)C (A, B; a, ). (z) r n=0 n+1 r=0 ¯ X X Next, instead of inverse factorials 1/(z)m we can utilize the asymptotically equivalent se- quence {1/(z+σ)m}m≥0 by employing the connection formula (see [28, (10)] or [29, (138.15)]) ∞ u k! ∞ v k! U(z)= k+1 = k+1 , (19) (z)k+1 (z + σ)k+1 Xk=0 Xk=0 with the coeﬃcients related by

k σ + j − 1 k (σ) v = u = j u . (20) k+1 j k+1−j j! k+1−j j=0 j=0 X X The inverse formula reads [38, 2.1(1)]

k σ u = (−1)j v . k+1 j k+1−j j=0 X Combining these facts with (16), we arrive at (Cj = Cj(A, B; a, )) ¯ n−j W (z) ∞ (−1)n n ∞ n! n (σ) (−1)n−j = (−1)js(n, j)C = j (−1)rs(n−j, r)C ν (z) j (z + σ) (n − j)!j! r n=0 n+1 j=0 n=0 n+1 j=0 r=0 X X X X X ∞ 1 n n−r n = (−1)rC (−1)n−j (σ) s(n − j, r) (z + σ) r j j n=0 n+1 r=0 j=0 X X X ∞ 1 n n n ∞ 1 n = (−1)rC (−1)k (σ) s(k,r)= C s (n, r) (z + σ) r k n−k (z + σ) r σ n=0 n+1 r=0 n=0 n+1 r=0 X X Xk=r X X which is precisely (18). To compute the convergence abscissa denote x = z + σ and consider the function p Γ(A x + a − A σ) Ω(x)= ρ−σW (z)= ρ−x k=1 k k k . q Γ(B x + b − B σ) Q j=1 j j j Denote a′ = a − σA,’ =-σB. Then, clearly,Qµ(a′, ’)= µ(a, ) = 1 (by the hypotheses of the ¯ ¯ ¯ theorem; µ(a, ) is deﬁned in 13). According to [17, Theorem¯ 6] ¯ 1 (B, ’) Ω(x)= tx−1Hp,0 ρt ¯ dt, q,p (A, a′) Z0

11 p,0 where Hq,p (ρt) is a particular case of Fox’s H-function deﬁned by c+i∞ p (B, ’) 1 Γ(Aks + ak − Akσ) Hp,0 ρt ¯ = k=1 (ρt)−sds q,p (A, a′) 2πi q Γ(B s + b − B σ) Z Q j=1 j j j c−i∞

(further details about the deﬁnition of Fox’s HQfunction can be found in [19, Chapter 1]; see also [17, 18]). By [19, Theorem 1.5] the following representation is true

p ∞ N −1 (B, ’) kl Hp,0 ρt ¯ = H (ρt)(ak −Akσ+l)/Ak logi(ρt), q,p (A, a′) kli k=1 l=0 i=0 X X X where the two outer summations run over all poles of the function Ω(x) and Nkl denotes the order of the pole at the point akl =(−ak + Akσ − l)/Ak. The explicit form of the constants Hkli is immaterial here. We emphasize, however, that if the pole of the numerator at x = akl cancels out with a pole of the denominator, then we put Nkl = 0 and the corresponding term is omitted from the above summation. Comparing this representation with (9) we conclude κ p,0 by Nørlund’s argument [28, p.339] explained below (9) that the order of Hq,p (ρt)/t on the circle |1 − t| = 1 is given by

κ = − min {ℜ(ak/Ak − σ − 1)} = max {ℜ(−ak/Ak)} + ℜ(σ)+1, k k where the minimum is taken over the indices k ∈ {1,...,p}, such that Ω(x) has a pole at xk = −ak/Ak + σ and −xk is not a nonnegative integer. Then by Theorem 3 the abscissa of convergence λΩ of Ω satisﬁes λΩ ≤ κ − 1 = maxk {ℜ(−ak/Ak)} + ℜ(σ) if κ − 1 ≤ 0 or − mink {ℜ(ak/Ak)}≤−ℜ(σ). The last condition can be interpreted as follows: the abscissa of the rightmost pole of W (z) does not exceed the abscissa of the rightmost pole of the series on the right hand side of (18). So that by the reverse change of variable z = x − σ the convergence abscissa λ of the original series (18) satisﬁes λ ≤ maxk {ℜ(−ak/Ak)}. If, on the contrary, κ − 1 > 0 or − mink {ℜ(ak/Ak)} > −ℜ(σ), then λΩ = κ − 1 = maxk {ℜ(−ak/Ak)}+ℜ(σ) implying that the convergence abscissa λ of the original series (18) is λ = maxk {ℜ(−ak/Ak)} (the abscissa of the rightmost pole of W (z)). Note, that for the rightmost pole xk∗ = −ak∗ /Ak∗ +σ the situation −xk∗ ∈ N0 contradicts − mink {ℜ(ak/Ak)} > −ℜ(σ), so that under this condition we necessarily have the equality λ = maxk {ℜ(−ak/Ak)}. It is easy to modify the above theorem to get rid of the restriction µ = 1. p q Corollary 1 Let Ak, Bj > 0 satisfy k=1 Ak = j=1 Bj and let θ, ak, bj, k = 1,...,p, j = 1,...,q, be arbitrary complex numbers. Suppose further that ν, ρ and µ are deﬁned in (12) and (13), respectively. Then P P p Γ(A z + a ) ∞ h Γ(z + θ + 1) W (z)= ρ−z k=1 k k = n , (21) q Γ(B z + b ) Γ(z + θ + µ + n + 1) j=1 j j n=0 Q X where the series converges inQ the half-plane ℜz > β with β equal to the real part of the rightmost pole of the function W (z)Γ(z + θ + µ)/Γ(z + θ +1). The coeﬃcients are computed by n (−1)kC (A, B; a, ) ′ ′ ′ r (n+µ) hn = ν Cr(A , B ; a , ’)sθ+µ(n, r)= νΓ(n + µ) ¯ B (−θ), ¯ k!Γ(r + µ) k r=0 X r+Xk=n 12 where A′ =(A, 1), B′ =(B, 1), a′ =(a, θ + µ), ’ =(,θ + 1) and C (A′, B′; a′, ’) are deﬁned ¯ r ¯ in (14). Moreover, if ℜ(β + θ + µ) > 0, then β is equal¯ to the convergence abscissa of the series (21).

Proof. If µ =6 1 we can take σ = θ + µ in Theorem 5 to get

p Γ(A z + a ) Γ(z + θ + µ) ∞ h W (z)= ρ−z k=1 k k = n , 1 q Γ(B z + b ) Γ(z + θ + 1) (z + θ + µ) j=1 j j n=0 n+1 Q X where h = ν n C (AQ′, B′; a′, ’)s (n, r) and the numbers C (A′, B′; a′, ’) are deﬁned n r=0 r ¯ θ+µ r ¯ in (14). Indeed, µ(a′, ’) = 1 by construction, so that Theorem 5 is applicable for W (z). ¯ 1 Multiplying bothP sides of (18) by Γ(z + θ +1)/Γ(z + θ + µ) we arrive at (21) together with the ﬁrst formula for hn and all claims regarding convergence. It remains to verify the second expression for hn. To this end employ the expansion [31, (43)]

1 ∞ (−1)kB(β+k)(−θ)(β) Γ(z + θ + 1) = k k , zβ k!Γ(z + θ + β + k + 1) Xk=0 where the series is known to converge for ℜz > 0. Applying Lemma 3 and making the necessary rearrangements we get

p Γ(A z + a ) ∞ C ∞ ∞ (−1)kB(r+µ+k)(−θ)(r + µ) Γ(z + θ + 1) ρ−z k=1 k k ∼ ν r = ν C k k q Γ(B z + b ) zr+µ r k!Γ(z + θ + r + µ + k + 1) j=1 j j r=0 r=0 Q X X Xk=0 ∞ k Q Γ(z + θ + 1)Γ(n + µ) (−1) Cr = ν B(n+µ)(−θ), Γ(z + θ + µ + n + 1) k!Γ(r + µ) k n=0 X r+Xk=n where this time Cr = Cr(A, B; a, ). As the inverse factorial series of a given function is unique whether it is convergent or¯ asymptotic we conclude that

k (−1) Cr(A, B; a, ) (n+µ) hn = νΓ(n + µ) ¯B (−θ). k!Γ(r + µ) k r+Xk=n To conclude this section we remark that the expansions presented in this note can proba- q p bly be generalized to the case when ∆ = j=1 Bj − k=1 Ak =0.6 For ∆ > 0 this expansion will be in terms of reciprocals of the gamma functions Γ(∆z +σ), while for ∆ < 0 the expan- sion will be in terms of Γ(−∆z + σ) (notP reciprocals!).P To derive such expansions one may apply the technique developed by Riney in [36], where the general ∆ > 0 case is deduced from the expansion similar to (21) for ∆ = 0. Riney’s results, however, are only asymptotic and pertain to the ”unweighted” case Ak = Bj = 1.

4. An identity for the Nørlund-Bernoulli polynomials. In this section we present an identity for the Nørlund-Bernoulli deﬁned by the generating function (17). Although its novelty is dubious, we could not locate it in the existing literature. The identity is obtained by comparing diﬀerent expansion for the simplest particular case of the function (2) and is given in the next proposition.

13 Theorem 6 The Nørlund-Bernoulli polynomials satisfy m−j (t − x + 1) m (x − t) m m B(t−x+m+1)(1 − x)= B(t−x+1)(t) j (−1)k s(m − k, j)(t) m! m j j! k k j=0 X Xk=0 and 1 m (1 − t − x) m t (t + x) s(m, j)B(t+x)(t) j = B(t+x+j)(x) j , m! j j! m − j j (j!)2 j=0 j=0 X X where s(m, j) is the Stirling number of the ﬁrst kind. Proof. We will use an asymptotic expansion in inverse powers of z for the ratio Γ(z + t)/Γ(z + x) found by Tricomi and Erd´elyi in [40]. This expansion is given on page 141 of [40], but we will rewrite it in a slightly diﬀerent form using the deﬁnition of the coeﬃcients n [40, (19)] and the identity Γ(1 − a)/Γ(1 − a − n)=(−1) (a)n: Γ(z + t) ∞ (−1)nB(t−x+1)(t)(x − t) ∼ n n as z →∞, (22) Γ(z + x)zt−x+1 n!zn+1 n=0 X where −x, −t, x − t∈ / N and z is in C cut along the ray connecting −α and −α −∞. Following [46], we can substitute the expansion [27, (6) on p. 78] 1 ∞ (−1)js(n + j, n) = zn+1 (z) j=0 n+j+1 X in (22) to get

Γ(z + t) ∞ (−1)nB(t−x+1)(t)(x − t) ∞ (−1)js(n + j, n) ∼ n n Γ(z + x)zt−x+1 n! (z) n=0 j=0 n+j+1 X X ∞ (−1)m m s(m, n)B(t−x+1)(t)(x − t) = n n . (23) (z) n! m=0 m+1 n=0 X X On the other hand Nørlund derived the expansion [31, (43)] Γ(z + t) ∞ (−1)mB(t−x+m+1)(1 − x)(t − x + 1) = m m (24) Γ(z + x)zt−x+1 m!(z + t) m=0 m+1 X convergent in some right half-plane. Equating the coeﬃcients in (23) and (24) we get an identity of the form (19) with (−1)mB(t−x+m+1)(1 − x)(t − x + 1) (−1)m m s(m, n)B(t−x+1)(t)(x − t) v = m m , u = n n . m+1 (m!)2 m+1 m! n! n=0 X After some rearrangement (20) takes the form

(t − x + 1) m m m−k (x − t) m B(t−x+m+1)(1 − x)= (−1)k (t) s(m − k, j)B(t−x+1)(t) j m! m k k j j! j=0 Xk=0 X m−j m (x − t) m = B(t−x+1)(t) j (−1)k s(m − k, j)(t) . (25) j j! k k j=0 X Xk=0

14 Applying the inversion formula below (20) and changing 1 − x 7→ x we ﬁnally obtain

1 m (1 − t − x) m t (t + x) s(m, j)B(t+x)(t) j = B(t+x+j)(x) j . m! j j! m − j j (j!)2 j=0 j=0 X X

Acknowledgements. We thank Gerg˝oNemes for contributing a lemma to the original version of this paper and useful discussions. This research was supported by the Russian Science Foundation under project 14-11-00022.

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18