Stirling Numbers and Inverse Factorial Series

December 28, 2020 Stirling numbers and inverse factorial series

Khristo N Boyadzhiev Department of Mathematics Ohio Northern University, Ada, Ohio 45810, USA [email protected]

Abstract We discuss inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such Stirling numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.

Key words: Inverse factorial series, Stirling numbers, polylogarithm, logarithm antiderivatives, harmonic numbers, digamma function, Nielsen’s beta function, incomplete gamma function. MSC: 11B73; 30B50; 33B30; 40A30; 65B10; 65D05

1. Introduction

n The unsigned Stirling numbers of the first kind  first appeared in James Stirling’s book k Methodus Differentialis (published in Latin) which appeared in 1730 (see [26] for a translation and comments). These numbers have numerous applications in combinatorics and analysis ([1, 3, 6, 9, 10, 12, 24]). In this paper we present various identities involving Stirling numbers of the first kind and inverse factorial series. These numbers are usually defined by their ordinary generating function

n n k (1) xx(+ 1)...( x +− n 1) =∑  x k =0 k

or by their exponential generating function

[−− ln(1xx )] kn∞ n (2) =∑  , |xk | <≥ 1, 0 kn!!nk= k

(see [9, 10]). However, there is another, less known generating function coming from the original works of James Stirling

11∞ n (3) k+1 = ∑  x nk= k xx(++ 1)...( x n )

in terms of inverse factorials (see [3] and [26, p.171]). Series of this form are called inverse factorial series. As shown by Milne-Thomson in Chapter 10 of his book [14], for appropriate functions fz() defined for Re(z )>>δ 0 expansions of the form

∞ a fz()= ∑ n n=0 zz(++ 1)...( z n )

are convergent and the coefficients are uniquely determined. See also Whittaker and Watson [30], pp. 142-144. Inverse factorial series were thoroughly studied by Norlund in [18, 19, 20], Obrechkoff [22], and Watson [27], and in more recent times by Wasow in [28, Chapter 11], Weniger [29], and Karp with Prilepkina [13]. Instead of listing general conditions for convergence, in many cases it is more practical to prove convergence by using the ratio test or the Raabe-Duhamel test. Formula (3) brings immediately to an important result. Setting zm= , an integer, summing for m =1,2,... and changing the order of summation we come to the representation

∞∞n 1 ζ (k += 1) ∑∑ nk=k m =1 mm(++ 1)...( m n )

where ζ ()s is Riemann’s zeta function

∞ 1 ζ ()ss= ∑ s ,Re()1> . n=1 n

It is easy to compute that

∞ 11 ∑ = m=1 mm(++ 1)...( m n ) nn !

(see, for example, [4]) so that

∞ n 1 (4) ζ (k += 1) ∑  . nk= k nn!

This result is not new, it was known to Charles Jordan at least in 1939 (see equation (6) on p. 166 and also equation (11) on p. 194 in his book [12]). Jordan even proved a more general formula ([12], p. 343]); for another proof see [4]. Informative comments on equation (4) can be found in Adamchik’s paper [1]. See also problems 46 and 47 in Chapter 1 of [24].

Equation (4) easily extends to the Hurwitz zeta function

∞ 1 ζ (,)sa =∑ s , Re (s )>> 1, a 0 n=0 ()na+ since from (3)

11∞ n k+1 = ∑  ()ma+ nk= k (mama+ )( ++ 1)...( man ++ ) and summing for m = 0,1,2,...and changing the order of summation we find

∞ n ∞ 1 ζ (ka+= 1, ) ∑ ∑ . nk= k m=0 (mama+ )( ++ 1)...( man ++ )

Evaluating the sum in the braces we come to

∞ n 1 (5) ζ (ka+=Γ 1, ) ( a )∑  nk= k nΓ+() na

which reduces to (4) for a =1. Another function “close” to Riemann’s zeta function is the polylogarithm

∞ xn Lis (x ) = ∑ s n=1 n

where Lin (1)= ζ (n ) . We are motivated now to ask the question: can equation (4) be extended to the polylogarithm? In the next section we give an affirmative answer in Theorem 3, which is our main result. Further in the paper we show various applications of the inverse factorial series. This material has a review character. More precisely, in Section 3 we provide several examples of inverse factorial series and in Section 4 we show how Stirling number identities can be used to generate new inverse factorial representations. In Section 5, we reprove the asymptotic expansion of Nielsen’s beta function β ()z and the incomplete gamma function γ (,zx ). In Section 6 we prove a formula relating a binomial sum to inverse factorial series with Stirling numbers.

2. Representation of the polylogarithm Let 11 HH=+++1 ... , = 0 n 2 n 0 be the harmonic numbers. First we have the following lemma.

Lemma 1. For any n ≥ 0

∞ pn+−n1 x (− 1) n (6) ∑ =+Pxn ( ) (1 −−x ) ln(1 x ) p=1 pp(++ 1)...( p n ) n! where Pxn () is the polynomial

nn−1 j nk− 1  nnnn(xx−− 1) 1 ( 1)  (7) Pxnn( )= Hx ( −+ 1) ∑∑ =Hxn( −+ 1)   n!! jk=01jknj−  n =  k 

1 with P (1) = . n nn!

The presence of the harmonic numbers Hn here is very interesting!

Proof. Let

∞ x pn+ fxn ()= ∑ p=1 pp(++ 1)...( p n )

for n = 0,1,... . Differentiating this function n times we find

n dx∞ p fn ()xx=∑ =−−ln(1 ) dx p=1 p

We see that fxn () is the n -th antiderivative of −−ln(1x ) satisfying the conditions (8) below. We will find a closed form evaluation for this series.

Clearly (d / dx ) fmm () x= f−1 () x . When n = 0 ,

∞ x p fx0 () =∑ =−−ln(1x ) p=1 p

and ff12, ,... fn are obtained from here by repeated integration. The constant of integration is taken zero each time so that according to (4) 111 1 (8) ff(1)= , (1) = , f (1) = , ..., f (1) = . 121!1 2!2 3 3!3 n nn! The process starts this way

=− − = − − = − − ++ f1( x )∫∫ ln(1 xdx ) ln(1 xd ) (1 x ) (1 x ) ln(1 x ) x C =−(1x ) ln(1 −+ xx ) then

31x fx() = x22 −−(1 − x ) ln(1 − x ) 2 4 22 etc. We evaluate each time at x =1 by taking limits,

lim(1−x )n ln(1 −= xn ) 0, = 1,2,... . x→1

Integrating repeatedly by parts we come to the representation

∞ pn+−n1 x (− 1) n (9) fnn() x=∑ =+Px( ) (1 −−x ) ln(1 x ) p=1 pp(++ 1)...( p n ) n! where Pxn () is a polynomial of degree n which we write for convenience in the form

nn−1 Pxnn( )= a ( x − 1) + a n−1 ( x − 1) ++ ... ax 10 ( −+ 1) a.

1 Obviously af=(1) = . To compute the other coefficients we repeatedly differentiate in (9) 0 n nn!

and set x →1. The coefficient an requires a longer (but elementary) computation. Namely,

differentiating equation (9) n times and setting x →1 brings to the equation

n n (− 1) k = + 0!nan ∑ k =1 k k and using the well-known identity

n n (− 1) k −1 = ∑ Hn . k =1 k k

we find that na! nn= H as desired. Thus we come to the representation (7) which completes the proof of the lemma. Setting x = −1 in (7) we compute that

(− 1)nn 2n 1 −= Pn ( 1) ∑ k nk!2k =1

and we come to the simple result: Corollary 2. For every n ≥ 0 we have

∞ (− 1)pn 2n 1 = − (10) ∑∑k ln 2 pk=11pp(++ 1)...( p n ) n != 2 k

The following theorem extends (4).

Theorem 3. For k ≥ 0 , ||1x < , we have

∞ n 1 (− 1) n−1 = + −−n (11) Likn+1 (x ) ∑  n Px( ) (1x ) ln(1 x ) nk= k xn!

where Pxn () is the polynomial from (7).

When we set x →1 in (11) the equation turns into (4). Proof. We write (3) in the form

11∞ n k+1 = ∑  p nk= k pp(++ 1)...( p n )

then multiply both sides by x p and sum for p from 1 to infinity. After changing the order of summation we write

∞∞n 1 x pn+ = Lik +1 (x ) ∑∑ n  nk= k xp=1 pp(++ 1)...( p n )

and the result follows from Lemma 1. Remark. A mentioned above, equation (9) represents the n th antiderivative of −−ln(1x ) with

the conditions (8). The functions fx2 () and fx3 () can be found in the table [21] of Prudnikov et al.; f2 is entry 5.2.6 (3) and f3 is entry 5.2.6 (12). They are listed also in Hansen’s table [11],

f2 is entry 5.7.40 and f3 is entry 5.6.29. The table [11] has some general formulas similar to (9) like 10.4.15 and 10.4.17. Hansen refers to the works of Schwatt (see [23], pp.191-192). The functions fn were studied also by Mathar [14] and a similar study can be found in [15].

We finish this section with a related result.

Proposition 4. For every k ≥1

k + 3 1 k −∞1 n ψ '(n ) ζ(k+ 2) −∑∑ ζζ ( j + 1) ( kj −+ 1) =  . 2 2!j=1 nk= k n

where ψ ()z=ΓΓ '()/() zz is the digamma function.

Proof. From (3) we have

∞ HHppn k+1 = ∑  p nk= k pp(++ 1)...( p n )

and then we sum for p =1,2,... . Interchanging the summations on the right hand side we write

∞∞∞ H p n H p ∑∑k+1 = ∑ . p=1 p nk= k p=1 pp(++ 1)...( p n )

It is known that for n ≥1

∞∞H 1 1ψ '(n ) p = = ∑∑2 pm=10pp(++ 1)...( p n ) n != ( n + m ) n !

(see, for instance, [4]). The value of the Euler sums on the left hand side is well-known

∞−H + 1 k 1 p =k 3 ζ + − ζζ + −+ ∑∑k +1 (k 2) ( j 1) ( kj 1) pj=11p 2 2 =

(see [1] or [22, p. 131]). This completes the proof.

3. Examples of inverse factorial expansions First we list several known series and then we add some new. Example 1. Using the properties nn++11  = n!, = nH! n 12 

we find from (3) with k =1and k = 2 the following two expansions

1!∞ n (12) = ∑ . z−1n=0 zz ( ++ 1)...( z n )

1 ∞ nH! = n (13) 2 ∑ (z− 1)n=0 zz ( ++ 1)...( z n )

by making in (3) the substitutions nn→+1, zz →− 1 .

Both series are convergent for Re(z )> 1. This convergence can be verified by the Raabe-

Duhamel test. For instance, if an is the general term of the series in (12) we have for real z >1,

a limnzn −=> 1 1. an+1

The Raabe-Duhamel test confirms also the convergence in the following several expansions. Let w be a complex number. A series extending (12) is

1∞ ww (+ 1)...( w +− n 1) (14) = ∑ zw−n=0 zz( ++ 1)...( zn ) convergent for Re(zw )> Re( ) (see [19, p. 222]).

Example 2. Another example in the same spirit is the series

1!∞ nn (15) = ∑ , Re(z )> 2 (z−− 1)( z 2)n=0 zz ( + 1)...( z + n )

(cf. entry 6.6.14 in Hansen’s table [11]). Example 3. Next (see 6.6.39 in [11]) we have

∞∞1n !1 ψ = = > (16) '()zz∑∑2 , Re() 0 nn=00(zn+ )= n ++ 1 zz ( 1)...( zn + )

where ψ ()z=ΓΓ '()/() zz is the digamma function.

Example 4. Consider the beta function

1 tez−−1 ∞ zt ∞ (− 1) n β ()z= dt = dt = , Re()z > 0 ∫∫−t ∑ . 00t+++11 en=0 nz

This function was studied by Niels Nielsen [17] and is often called Nielsen’s beta function. For some properties and important integrals related to β ()z see [8].

Integration by parts gives

111 t z β ()z = + ∫ dt 2zzt0 (+ 1)

and repeating this again and again we come to the expansion

∞ n!1 β = (17) ()z ∑ n+1 . n=0 2zz (++ 1)...( z n )

This representation appears on p. 81 in Nielsen’s book [17]. It appears also on p. 352 in Norlund’s paper [18]. Example 5. Consider the lower incomplete gamma function

x γ (z , x )=∫ tzt−−1 e dt , x >> 0, Re ( z ) 0 . 0

We have (Temme [25, p. 279])

∞ x n γ (,z x )= xezx− ∑ . n=0 zz(++ 1)...( z n )

Convergence follows from the ratio test. Example 6. Jacques Binet proved the expansion for the log-gamma function

1 ∞ a lnΓ (z ) =z − ln zz −+ ln 2π +∑ n 2 n=1 (z++ 1)) z 2)...( zn + ) where

1 1 =1 + +− an ∫t − t( t 1)...( t n 1) dt . n 0 2

We can write this in the form (with a0 = 0 )

1Γ ()z 1 ∞ a ln −1− lnz += 1 ∑ n z 2π 2z n=0 zz(++ 1)( z 2)...( z + n ) (see Whittaker and Watson [30, p. 253]). Inverse factorial series appeared also in the works of Ramanujan, as discussed by Berndt in [2]. More details will be given in Section 5.

4. Generating new inverse factorial representations The following proposition makes it possible to generate more inverse factorial representations.

Proposition 5. Let aa01, ,..., an ,... be a sequence. Then we formally have

∞∞a 1 n n k = (18) ∑∑k +1  ∑ak . kn=00z= zz(++ 1)...( z n ) k=0k

For the proof we multiply both sides in (3) by a k , sum for k from zero to infinity and change the order of summation. We do not discuss convergence in the general case. Convergence can be checked easily in all particular examples below. The formula can be written also in the form Using the Stirling sequence transformation

n n bank= ∑  k =0 k

we can generate various inverse factorial representations. A short table of Stirling transform identities can be found in the Appendix in [5]. The entries in this table are written in terms of the (signed) Stirling numbers of the first kind snk(, ) for which

n nk− (19) =( − 1)snk ( , ) k

and in terms of snk(, ) formula (18) can be written in the form

∞∞(− 1) k a (− 1) n n k = (20) ∑∑k +1  ∑snk(, ) ak kn=00z= zz(++ 1)...( z n )  k= 0

k replacing ak by (− 1) ak and using (19).

The representations (12) and (13) follow from the identities

nnnn  ∑∑= n!,  k= nH! n . kk=00kk= 

Example 7. In the same line, using the identity

n n 2 2 (2) ∑ k= nH!( n +− Hnn H ) k =0 k

11 where H (2) =+1 ++ ... , we find from (18) the representation n 222n

∞∞kn2 ! = +−2 (2) ∑∑k +1 (HHHn nn) . kn=00z= zz(++ 1)...( z n )

Computing the series on the left hand side for z >1 we come to the inverse factorial series

zn+1!∞ = +−2 (2) 3 ∑ (HHHn nn) . (z− 1)n=0 zz ( ++ 1)...( z n )

Example 8. In this example we use the Stirling numbers of the second kind Snm(, ) which can be defined by the generating function

∞ xm (21) ∑ Snm(, ) xn = . n=0 (1−−x )(1 2 x )...(1 − mx )

For any two integers 0 ≤≤pn the following identity is true (entry (A30) in [5])

n nnn!   ∑ Sk(+ 1, p += 1) . k =0 kpp! 

According to Proposition 5 this implies the series identity

∞∞Sk(++ 1, p 1) 1 n! n = (22) ∑∑k +1 . kn=00z p!= zz (++ 1)...( z n ) p

In view of (21) with xz=1/ we find the representation

1 1!∞ n n (23) = ∑  (z− 1)( z − 2)...( z −− p 1) p !n=0 zz ( + 1)...( z + n ) p

convergent for Re(zp )>+ 1. For p = 0 this is equation (12) and for p =1 this is (15).

Example 9. Let dn be the sequence of Cauchy numbers of the second kind defined by the generating function

−xx∞ n = ∑ dn (1−−xx ) ln (1 )n=0 n !

(see [9, p. 293]). The numbers dn for n = 0,1,..., satisfy the identity (entry (A42) in [5])

n n 1 n ∑  =( − 1) dn k =0 k k +1

and from Proposition 5 it follows that

∞∞1 (− 1) n d = n > ∑∑k +1 ,1z . kn=00(k+ 1) z= zz ( ++ 1)...( z n )

The series on the left is easy to recognize and so we have the representation

1 ∞ (− 1) n d (24) −ln 1 −=∑ n ,z > 1. zn=0 zz(++ 1)...( z n )

It is interesting to compare this to the representation

1 ∞ (− 1) n c (25) ln 1+=∑ n ,z > 0 zn=0 zz(++ 1)...( z n )

where cn are the Cauchy numbers of the first kind defined by

xx∞ n = ∑ cn ln(1+ xn )n=0 !

(see [6, 9] , and [30, p. 144]). A similar representation can be found on p. 244 in [20].

5. Asymptotic expansions

We can use Proposition 5 to obtain asymptotic series on the powers of z−−n 1 for functions with inverse factorial representations. We will use Proposition 5 to reprove the asymptotic representation of Nielsen’s beta function β ()z from Example 4, the incomplete gamma function γ (,zx ), and ψ '(z ) . We can use either equation (18) or equation (20).

Setting

∞∞n!1 1n n β = = ()za∑n+1 ∑∑ k n=02zz (++ 1)...( z n ) nk= 00 zz ( ++ 1)...( z n ) = k

we solve for ak from the equation

n n n! = ∑ ak n+1 . k =0 k 2

Using the formulas for the Stirling transform of sequences (see the Appendix in [5]) we have

nn(− 1)k 1 k −=n = 1 (1)an ∑∑ Snkk (, )! k +1 Snkk(, )!− . kk=0022= 2 At this point we involve the geometric polynomials

n k (26) ωn ()x= ∑ Snkk (, )! x k =0

introduced and studied in [7]. It is known that

1 2 n+1 ω − =−=(1 2 )BE+ (0) n2 n +1 nn1

where Bk are the Bernoulli numbers and Exk () are the Euler polynomials. For the first equality see the solution of problem 6.76 on p. 559 in [10]. For the second equality see [3]. This way we have

(−− 1) nn( 1) a=−=(1 2n+1 ) BE(0) nn +12 nn+1 and we come to following proposition. Proposition 6. Nielsen’s beta function has the asymptotic series

∞∞(−− 1)nn (1 2+1 ) B 11 1 β = n+1 = − n (27) ()zE∑∑nn++11( 1)n (0) . nn=00nz+12= z

(Cf. Example 1 on p. 145 in [2]). In the same way we can use the representation of the

incomplete gamma function γ (,zx ) from Example 5. Solving for the coefficients an from the equation

n n n (28) ∑ axk = k =0 k we find that

n n kk (29) an =−−( 1)∑ Snk ( , )( 1) x . k =0

This time we use the exponential polynomials

n k ϕn ()x= ∑ Snk (, ) x k =0

(see [3, 7]). These polynomials appeared in the works of Ramanujan [2], E.T. Bell, J. Touchard, Gian-Carlo Rota, and many others. As mentioned in [3], these polynomials were used as early as 1843 by the prominent German mathematician Johann A. Grunert.

n Equation (29) says that axnn=−−( 1)ϕ ( ) , so that we have

∞ 1 γϕ=zx− −− n (z , x ) xe ∑ ( 1)n (x ) n+1 n=0 z

in accordance with Proposition 5.1 in [7] proved there by a different method. We notice that equations (28) and (29) in view of Proposition 5 lead to the interesting formula

∞∞xn 1 =−−nϕ (30) ∑∑( 1)n (x ) k +1 nk=00zz(++ 1)...( z n ) = z

which Ramanujan’s Example (c) on p.145 in [2] (see also p. 47 in [2]). In a similar manner, using the representation from Example 3

∞∞nn! 1 (−− 1)nn ! ( 1) ψ '(z ) = ∑∑= nn=00n+++1 zz ( 1)...( z n )= n + 1 zz ( ++ 1)...( z n )

we compare to (20) and solve for ak from the equation

(− 1)n n ! n = ∑ snk(, ) ak n +1 k =0

to find

n (− 1)k k ! an = ∑ Snk(, ) k =0 k +1

which are exactly the Bernoulli numbers [3], aBbn= . In view of (20) we find a new proof of the asymptotic expansion

∞ (− 1) k B ψ = k (31) '(z ) ∑ k =1 . k =0 z

6. A binomial formula with inverse factorials In this short section we prove a formula relating a binomial sum and an inverse factorial series with Stirling numbers of the first kind. Proposition 7. For any three integers pk≥≥1, 0, m ≥ 0 we have

m mn  (− 1) j ∞  1 = − (32) ∑∑ k +1 (p 1)!  . jn=00jk ()j+ p = nnm!(++ 1)( nm ++ 2)...( nmp ++ )

Proof . We start with the (exponential) generating function for the unsigned Stirling numbers of the first kind

∞ n kk n x (− 1) ln (1 −=xk ) !∑  n=0 k n!

where ||1x < . We multiply both sides by xxmp(1− ) −1 and integrate between 0 and 1 to get

11∞ kmpk−11n 1 nmp+− (− 1)∫∫x (1 − x ) ln (1 −= x ) dx k !∑  x(1− x ) dx . 00n=0 k n!

In the integral on the left hand side we make the substitution 1−=xe−t

1 ∞ (− 1)k∫∫x m (1 − x ) p−1 ln k (1 −=− x ) dx (1 e−t ) m e −+ pt t t k e − t dt 00

mmmm∞  1 =−=−( 1) jt k e−+() j pt dt k! ( 1) j . ∑∑ ∫  k +1 jj=00jj0 =  ()jp+

For the integral on the right hand side we use the representation

1 (p − 1)! ∫ xnm+−(1−= x ) p1 dx 0 (nm++ 1)( nm ++ 2)...( nmp ++ ) which comes from Euler’s beta function

1 ΓΓ()uv () B( u , v )=−=∫ xuv−−11 (1 x ) dx . 0 Γ+()uv

Equating both sides we come to the desired formula. Example 10. In particular, for p =1 in the above formula we have

m mn  (− 1) j ∞  1 = (33) ∑∑ k +1  . jn=00jk (j+ 1) = nn!(++ m 1)

For m = 0, m =1, and m = 2 this gives correspondingly

∞ n 1 1 = ∑  n=0 k nn!(+ 1)

11∞ n −= 1 k +1 ∑  2 n=0 k nn!(+ 2)

11∞ n 1 −+ = 1 kk+1 ∑  2 3 n=0 k nn!(+ 3) etc.

References [1] Victor Adamchik, On Stirling numbers and Euler sums, J. Comput. Applied Math. 79 (1997) 119-130. [2] Bruce C. Berndt, Ramanujan’s notebooks, Part 1. Springer 1983. [3] Khristo N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag., 85, No. 4, (2012), 252-266. [4] Khristo N. Boyadzhiev Evaluation of some simple Euler-type series, Far East J. Math. Sci. (FJMS), 29 (1) (2008), 1 -11. [5] Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, World Scientific, 2018. [6] Khristo N. Boyadzhiev , New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials (submitted). Also https://arxiv.org/abs/1911.00186 [7] Khristo N. Boyadzhiev , A Series transformation formula and related polynomials, Int. J. Math. Math. Sci., Vol. 2005 (2005), Issue 23, 3849-3866.

[8] Khristo N. Boyadzhiev, Louis Medina, and Victor Moll, The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function, SCIENTIA: Series A: Mathematical Sciences, 18 (2009), 61-75. [9] Louis Comtet, Advanced Combinatorics, Kluwer, Dordrecht, 1974. [10] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, New York, 1994. [11] Eldon R. Hansen, A Table of Series and Products, Prentice Hall, 1975. [12] Charles Jordan, Calculus of Finite Differences, Chelsea, New York, 1950 (First edition: Budapest 1939). [13] Dmitri B. Karp, Elena G. Prilepkina, An inverse factorial series for a general gamma ratio and related properties of the Norlund-Bernoulli polynomials, J. Math. Sci., 234 (2018), 680–696. [14] Richard J. Mathar. Series of reciprocal powers of k-almost primes; available at https://arxiv.org/abs/0803.0900v3 [15] Luis A Medina, Victor Moll, and Eric Rowland, Iterated primitives of logarithmic powers, Int. J. Number Theory, 7(3) (2011), 623-634. [16] Louis M. Milne-Thomson, The Calculus of Finite Differences, MacMillan, London, 1951. [17] Niels Nielsen, Die Gammafunktion, Chelsea, 1965 (First published 1906.) [18] Niels E. Nørlund, Sur les séries de facultes, Acta Math., 37 (1914), 327-387. [19] Niels E. Nørlund, Leçons Sur les Séries d'Interpolation, Gauthier-Villars, 1926. [20] Niels E. Nørlund, Vorlesungen Uber Differenzenrechnung, Springer, 1954 [21] Anatoli P. Prudnikov, Yury A. Brychkov, Oleg I. Marichev, Integrals and Series, vol.1, Elementary Functions, Gordon and Breach 1986. [22] Nikola D. Obrechkoff, Summation by Euler's transform of the series of Dirichlet, factorial series, and the series of Newton, Pliska Stud. Math., 28 (2017), 7–127. (Republication, first published in 1939; Addendum by Peter Russev, ibid. 128-132.) [23] Isaak J. Schwatt, An Introduction to the Operations with Series, Chelsea, New York, 1924. [24] Hary M. Srivastava and Junesang Choi, Series Associated with Zeta and Related Functions, Kluwer, 2001. [25] Nico M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 1996.

[26] Ian Tweddle, James Stirling’s Methodus Differentialis: An Annotated Translation of Stirling’s Text, Springer, New York, 2003. [27] George N. Watson, The transformation of an asymptotic series into a convergent series of inverse factorials, Rend, Circ, Mat, Palermo, 34 (1912), 41-88. [28] Wolfgang R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover, 2002. [29] Ernst J. Weniger, Summation of divergent power series by means of factorial series, Appl. Numerical Math., 60 (12) (2010), 1429-1441. [30] Edmund T. Whittaker and George N. Watson, A Course of Modern Analysis, Cambridge University Press, 1992.