The function _{v}M_{m}(s; a, z) and some well-known sequences

Aleksandar Petojević

Aleja Maršala Tita 4/2 24000 Subotica Yugoslavia

Email address: apetoje@ptt.yu

Abstract

In this paper we define the function_{v}M_{m}(s; a, z), and we study the special cases_{1}M_{m}(s; a, z) and _{n}M_{−1}(1; 1, n + 1). We prove some new equivalents of Kurepa’s hypothesis for the left factorial. Also, we present a generalization of the alternating factorial numbers.

1 Introduction

Studying the Kurepa function

K(z) = !z = ∫_{0}^{+∞} \frac{t^{z} − 1}{t − 1} e^{−t} dt (Re z > 0),

G. V. Milovanović gave a generalization of the function

M_{m}(z) = ∫_{0}^{+∞} \frac{t^{z+m} − Q_{m}(t; z)}{(t − 1)^{m+1}}e^{−t}dt (Re z > −(m + 1)),

where the polynomials Q_{m}(t; z), m = −1, 0, 1, 2, ... are given by

Q_{−1}(t; z) = 0 Q_{m}(t; z) = ∑_{k=0}^{m} \binom{m + z}{k}(t − 1)^{k}.

This work was supported in part by the Serbian Ministry of Science, Technology and Development under Grant # 2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods.

The function {M_{m}(z)}_{m=−1}^{+∞} has the integral representation

M_{m}(z) = \frac{z(z + 1) ⋯ (z + m)}{m!} ∫_{0}^{1} ξ^{z−1}(1 − ξ)^{m}e^{(1−ξ)/ξ}Γ(z, \frac{1 − ξ}{ξ} ) dξ,

where Γ(z, x), the incomplete gamma function, is defined by

Γ(z, x) = ∫_{x}^{+∞} t^{z−1}e^{−t} dt.  (1)

Special cases include

M_{−1}(z) = Γ(z) and M_{0}(z) = K(z),  (2)

where Γ(z) is the gamma function

Γ(z) = ∫_{0}^{+∞} t^{z−1}e^{−t} dt.

The numbers M_{m}(n) were introduced by Milovanović [10] and Milovanović and Petojević [11]. For non-negative integers n, m ∈ ℕ the following identities hold:

M_{m}(0) = 0, M_{m}(n) = ∑_{i=0}^{n−1} \frac{(−1)^{i}}{i!} ∑_{k=i}^{n−1} k!\binom{m + n}{k + m + 1}.

For the numbers M_{m}(n) the following relations hold:

M_{m}(n + 1) = n! + ∑_{ν=0}^{m} M_{ν}(n),

lim_{n→+∞} \frac{M_{ν}(n)}{M_{ν−1}(n)} = 1,

lim_{n→+∞} \frac{M_{m}(n)}{(n − 1)!} = 1.

The generating function of the numbers {M_{m}(n)}_{n=0}^{+∞} is given by

\frac{1}{m!} (A_{m}(x)e^{x−1} (Ei (1) − Ei (1 − x) + B_{m}(x)e^{x} − C_{m}(x))) = ∑_{n=0}^{+∞} M_{m}(n) \frac{x^{n}}{n!}

where A_{m}(x), B_{m}(x), and C_{m}(x) are polynomials defined as follows:

\frac{A_{m}(x)}{m!} = ∑_{k=0}^{m} \binom{m}{k}\frac{(x − 1)^{k}}{k!} ,

\frac{B_{m}(x)}{m!} = ∑_{ν=0}^{m−1} (∑_{k=1}^{m−ν} \binom{m}{k + ν}\frac{(−1)^{k−1}}{k} ∑_{j=0}^{k−1} \frac{(−1)^{j}}{j!} ) \frac{(x − 1)^{ν}}{ν!} ,

\frac{C_{m}(x)}{m!} = ∑_{j=0}^{m−1} (∑_{ν=0}^{j}(−1)^{ν}\binom{j}{ν} ∑_{k=j+1}^{m} \frac{(−1)^{k−1}}{k − ν} \binom{m}{k}) \frac{(x − 1)^{j}}{j!} ,

Here Ei (x) is the exponential integral defined by

Ei (x) = p.v. ∫_{−∞}^{x} \frac{e^{t}}{t} dt (x > 0).  (3)

In this paper, we give a generalization of the function M_{m}(z) which we denote as vM_{m}(s; a, z). These generalization are of interest because its special cases include:

1M_{1}(1; 1, n + 1) = n!,
1M_{0}(1; 1, n) = !n,
nM_{−1}(1; 1, n + 1) = A_{n}.

where n!, !n, and A_{n} are the right factorial numbers (sequence A000142 in [17]), the left factorial numbers (sequence A003422 in [17]) and the alternating factorial numbers (sequence A005165 in [17]), respectively. They are defined as follows:

0! = 1, n! = n · (n − 1)!;  !0 = 0, !n = ∑_{k=0}^{n−1} k! and A_{n} = ∑_{k=1}^{n}(−1)^{n−k}k!. (4)

2 Definitions

We now introduce a generalization of the function M_{m}(z).

Definition 1 For m = −1, 0, 1, 2, ..., and Re z > v − m − 2 the function _{v}M_{m}(s; a, z) is defined by

vM_{m}(s; a, z) =

= ∑_{k=1}^{v} \frac{(−1)^{2v+1−k}Γ(m + z + 2 − k)}{Γ(z + 1 − k)Γ(m + 2)} L[s; _{2}F_{1}(a, k − z, m + 2, 1 − t)],

where v is a positive integer, and s, a, z are complex variables.

The hypergeometric function _{2}F_{1}(a, b; c, x) is defined by the series

2F_{1}(a, b, c; x) = ∑_{n=0}^{∞} \frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{x^{n}}{n!} (|x| < 1),

and has the integral representation

2F_{1}(a, b, c; x) = \frac{Γ(c)}{Γ(b)Γ(c − b)} ∫_{0}^{1} t^{b−1}(1 − t)^{c−b−1}(1 − tx)^{−a}dt,

in the x plane cut along the real axis from 1 to ∞, if Re c > Re b > 0. The symbols (z)_{n} and L[s; F (t)] represent the Pochhammer symbol

(z)_{n} = \frac{Γ(z + n)}{Γ(z)},

and Laplace transform

L[s; F (t)] = ∫_{0}^{∞} e^{−st}F (t)dt.

Table 1: The numbers _{1}M_{m}(1; a, n) for m = 1, 2, 3, 4 and a = 0, 1, 2
a = 0  in [17]  a = 1  in [17]  a = 2  in [17]
1M_{1}(1, 0, n)  A000217  _{1}M_{1}(1, 1, n)  A014144  _{1}M_{1}(1, 2, n)  A007489
1M_{2}(1, 0, n)  A000292  _{1}M_{2}(1, 1, n)  unlisted  _{1}M_{2}(1, 2, n)  A014145
1M_{3}(1, 0, n)  A000332  _{1}M_{3}(1, 1, n)  unlisted  _{1}M_{3}(1, 2, n)  unlisted
1M_{4}(1, 0, n)  A000389  _{1}M_{4}(1, 1, n)  unlisted  _{1}M_{4}(1, 2, n)  unlisted

The term “unlisted” in Table 1 means that the sequence cannot currently be found in Sloane’s on-line encyclopedia of integer sequences[17].

Lemma 1 Let m = −1, 0, 1, 2, …. Then

1M_{m}(1; 1, z) = M_{m}(z).

Proof. The proof presented here is due to Professor G. V. Milovanović. Since

(k + m + 1)! = (m + 1)!(m + 2)_{k} and (1 − z)_{k} = \frac{(−1)^{k}Γ(z)}{Γ(z − k)},

we have

\binom{m + z}{k + m + 1} = \frac{Γ(m + z + 1)}{Γ(z − k)(k + m + 1)!} = \frac{Γ(m + z + 1)}{Γ(z)(m + 1)!} · \frac{(1 − z)_{k}(−1)^{k}}{(m + 2)_{k}} ,

so that

\frac{t^{z+m} − Q_{m}(t; z)}{(t − 1)^{m+1}} = ∑_{k=0}^{+∞} \binom{m + z}{k + m + 1}(t − 1)^{k}  (|t − 1| < 1)

= \frac{Γ(m + z + 1)}{Γ(z)(m + 1)!} ∑_{k=0}^{+∞} \frac{(1 − z)_{k}(1)_{k}}{(m + 2)_{k}} · \frac{(1 − t)^{k}}{k!}

= \frac{Γ(m + z + 1)}{Γ(z)(m + 1)!} _{2}F_{1}(1, 1 − z, m + 2; 1 − t).

3 The function _{1}M_{m}(s; a, z)

3.1 The numbers {_{1}M_{m}(1; −n, r)}_{r=0}^{+∞} _{n=0}^{+∞} _{m=−1}^{+∞}

We have

1M_{m}(1; −n, z) = \frac{Γ(m + z + 1)}{Γ(z)Γ(m + 2)} L[1; _{2}F_{1}(−n, 1 − z, m + 2; 1 − t)],

starting with the polynomials

∑_{k=0}^{∞} \binom{m + z}{k + m + 1}\binom{n}{k}(1 − t)^{k}, n ∈ ℕ.

Since

(−n)_{k} = (−1)^{k} \frac{n!}{(n − k)!}

we have

∑_{k=0}^{∞} \binom{m + z}{k + m + 1}\binom{n}{k}(1 − t)^{k} = \frac{Γ(m + z + 1)}{Γ(z)Γ(m + 2)} _{2}F_{1}(−n, 1 − z, m + 2; 1 − t)  (5)

or, by continuation,

= π cosec πz ∫_{0}^{1} ξ^{1−z}(1 − ξ)^{m+z+1}(1 − (1 − t)ξ)^{n}dξ.

Hence, the following definition is reasonable. Definition 2 For n ∈ ℕ and m = −1, 0, 1, 2, ..., the polynomials z ↦ _{m}P_{n}(z) are defined by

mP_{n}(z) = L[1; _{2}F_{1}(−n, 1 − z, m + 2; 1 − t)].

Table 2: The polynomials _{m}P_{2}(z), _{m}P_{3}(z) and _{m}P_{4}(z)
m  _{m}P_{2}(z)  _{m}P_{3}(z)  _{m}P_{4}(z)
−1  \frac{1}{2}z^{2} − \frac{3}{2}z + 2  −\frac{1}{3}z^{3} + \frac{7}{2}z^{2} − \frac{49}{6}z + 6  \frac{3}{8}z^{4} − \frac{61}{12}z^{3} + \frac{193}{8}z^{2}−
0  \frac{1}{6}z^{2} − \frac{1}{2}z + \frac{4}{3}  −\frac{1}{12}z^{3} + z^{2} − \frac{29}{12}z + \frac{5}{2}  −\frac{509}{12}z + 24
1  \frac{1}{12}z^{2} − \frac{1}{4}z + \frac{7}{6}  −\frac{1}{30}z^{3} + \frac{9}{20}z^{2} − \frac{67}{60}z + \frac{17}{10}

The polynomials_{m}P_{n}(z) can be expressed in terms of the derangement numbers (sequence A000166 in [17])

S_{k} = k! ∑_{ν=0}^{k} \frac{(−1)^{ν}}{ν!} (k ≥ 0)

in the form Theorem 1 For m = −1, 0, 1, 2, ... and n ∈ ℕ we have

mP_{n}(z) = \binom{n + m + 1}{n}^{−1} ∑_{k=0}^{∞} \binom{n + m + 1}{k + m + 1}\binom{z − 1}{k}(−1)^{k}S_{k}.

Proof. Using the relation (5) we have

mP_{n}(z) = \frac{Γ(z)Γ(m + 2)}{Γ(m + z + 1)} ∫_{0}^{∞} e^{−t} ∑_{k=0}^{∞} \binom{m + z}{k + m + 1}\binom{n}{k}(1 − t)^{k}dt

= \frac{Γ(z)(m + 1)!}{Γ(m + z + 1)} ∑_{k=0}^{∞} \frac{Γ(m + z + 1)}{Γ(z − k)(k + m + 1)!}\binom{n}{k} ∫_{0}^{∞} e^{−t}(1 − t)^{k}dt

= \binom{n + m + 1}{n}^{−1} ∑_{k=0}^{∞} \binom{n + m + 1}{k + m + 1}\binom{z − 1}{k} ∫_{0}^{∞} e^{−t}(1 − t)^{k}dt.

Now use

L[s; (t + α)^{z−1}] = \frac{e^{αs}Γ(z, αs)}{s^{z}} (Re s > 0),

to obtain

mP_{n}(z) = \binom{n + m + 1}{n}^{−1} ∑_{k=0}^{∞} \binom{n + m + 1}{k + m + 1}\binom{z − 1}{k}(−1)^{k} \frac{Γ(k + 1, −1)}{e}.

Here Γ(z, x) is the incomplete gamma function. The result follows from

Γ(k + 1, −1) = eS_{k}.

Lemma 2 For n ∈ ℕ we have

−1P_{n}(z) = 1 + n! ∑_{k=1}^{n} \frac{(−1)^{k}S_{k}}{(n − k)! (k!)^{2}} ∏_{i=1}^{k}(z − i).

Proof. Applying Theorem 1 for m = −1, we have

−1P_{n}(z) = ∑_{k=0}^{n} \binom{n}{k}\binom{z − 1}{k}(−1)^{k}S_{k},

= 1 + n! ∑_{k=1}^{n} \frac{(−1)^{k}S_{k}}{(n − k)!(k!)^{2}} ∏_{i=1}^{k}(z − i).

Remark 1 Let the sequence X_{n,k} be defined by

X_{n,k} = \begin{cases}Y_{n}, & if n = k, \\ (n − k)X_{n−1,k}, & if n > k, \end{cases}

where Y_{n} = (n!)^{2}. Since X_{n,k} = (n − k)! (k!)^{2} we have

−1P_{n}(z) = 1 + n! ∑_{k=1}^{n} \frac{(−1)^{k}S_{k}}{X_{n,k}} ∏_{i=1}^{k}(z − i).

Theorem 2 For m = −1, 0, 1, 2, ... and n ∈ ℕ we have

mP_{0}(z) = _{m}P_{1}(z) = 1
mP_{n}(z) = \frac{1}{n + m + 1} [(m + 1)_{m−1}P_{n}(z) + n_{m}P_{n−1}(z)], m > −1.

Proof. According to Theorem 1 we have

\frac{1}{n + m + 1} [(m + 1)_{m−1}P_{n}(z) + n_{m}P_{n−1}(z)] =

\frac{(m + 1)!n!}{(n + m + 1)!} ∑_{k=0}^{∞} \binom{n + m}{k + m}\binom{z − 1}{k}(−1)^{k}S_{k} +

+ \frac{(m + 1)!n!}{(n + m + 1)!} ∑_{k=0}^{∞} \binom{n + m}{k + m + 1}\binom{z − 1}{k}(−1)^{k}S_{k}.

The recurrence for _{m}P_{n}(z) now follows from

\binom{a}{b} + \binom{a}{b + 1} = \binom{a + 1}{b + 1}.

Corollary 1 For m = −1, 0, 1, 2, ... and n ∈ ℕ we have

1M_{m}(1; −n, z) = \frac{n! z(z + 1) … (z + m)}{(n + m + 1)!} ∑_{k=0}^{∞} \binom{n + m + 1}{k + m + 1}\binom{z − 1}{k}(−1)^{k}S_{k}.

Corollary 2 For n ∈ ℕ the result is as follows

1M_{−1}(1; −n, z) = 1 + n! ∑_{k=1}^{n} \frac{(−1)^{k}S_{k}}{(n − k)! (k!)^{2}} ∏_{i=1}^{k}(z − i),

1M_{m}(1; 0, z) = _{1}M_{m}(1; −1, z) = \frac{Γ(m + z + 1)}{Γ(z)Γ(m + 2)},

1M_{m}(1; −n, z) = \frac{1}{n + m + 1} · [ (m + z) · _{1}M_{m−1}(1; −n, z) +
+ n · _{1}M_{m}(1; −n + 1, z)], m > −1.

The numbers _{1}M_{m}(1; −n, r)_{r=0}^{∞} can now be evaluated recursively.

Table 3: The numbers _{1}M_{m}(1; −n, r) for m = 0, 1, 2, 3, 4 and n = 1, 2
1M_{m}(1, −n, r)  sequence in [17]  _{1}M_{m}(1, −n, r)  sequence in [17]
1M_{0}(1, −1, r)  0, 1, 2, 3, 4...  _{1}M_{0}(1, −2, r)  A000125
1M_{1}(1, −1, r)  A000217  _{1}M_{1}(1, −2, r)  A055795
1M_{2}(1, −1, r)  A000292  _{1}M_{2}(1, −2, r)  A027660
1M_{3}(1, −1, r)  A000332  _{1}M_{3}(1, −2, r)  A055796
1M_{4}(1, −1, r)  A000389  _{1}M_{4}(1, −2, r)  A055797

3.2 The numbers {_{1}M_{m}(1/n; m + 2, r)}_{r=0}^{+∞} _{n=1}^{+∞} _{m=−1}^{+∞}

In Table 4 twelve well-known sequences from [17] are given. These sequences are special cases of the function _{v}M_{m}(s; a, r) for v = 1, s = 1/n, and a = m + 2. The sequences have the following common characteristic.

Lemma 3 For m = −1, 0, 1, 2, ..., we have

1M_{m}(1/n; m + 2, z) = \frac{n^{z}Γ(z + m + 1)}{(m + 1)!}.

Proof. Since

2F_{1}(m + 2, 1 − z, m + 2, 1 − t) = t^{z−1}

we have

L[1/n; _{2}F_{1}(m + 2, 1 − z, m + 2, 1 − t)] = n^{z}Γ(z).

Table 4: The numbers _{1}M_{m}(1/n; m + 2, r) for m = −1, 0, 1, 2, 3, 4 and n = 1, 2, 3, 4
1M_{m}(1/n, m + 2, r)  in [17]  _{1}M_{m}(1/n, m + 2, r)  in [17]
1M_{−1}(1, 1, r)  A000142  _{1}M_{−1}(1/2, 1, r)  A066318
1M_{−1}(1/3, 1, r)  A032179  _{1}M_{−1}(1/4, 1, r)  unlisted
1M_{0}(1, 2, r)  A000142  _{1}M_{0}(1/2, 2, r)  A000165
1M_{0}(1/3, 2, r)  A032031  _{1}M_{0}(1/4, 2, r)  A047053
1M_{1}(1, 3, r)  A001710  _{1}M_{1}(1/2, 3, r)  A014297
1M_{1}(1/3, 3, r)  unlisted  _{1}M_{1}(1/4, 3, r)  unlisted
1M_{2}(1, 4, r)  A001715  _{1}M_{2}(1/2, 4, r)  unlisted
1M_{2}(1/3, 4, r)  unlisted  _{1}M_{2}(1/4, 4, r)  unlisted
1M_{3}(1, 5, r)  A001720  _{1}M_{3}(1/2, 5, r)  unlisted
1M_{3}(1/3, 5, r)  unlisted  _{1}M_{3}(1/4, 5, r)  unlisted
1M_{4}(1, 6, r)  A001725  _{1}M_{4}(1/2, 6, r)  unlisted
1M_{4}(1/3, 6, r)  unlisted  _{1}M_{4}(1/4, 6, r)  unlisted

3.3 Some equivalents of Kurepa’s hypothesis

The special values M_{−1}(z) = Γ(z) and M_{0}(z) = K(z) given in (2) yield

1M_{−1}(1, 1, n + 1) = n! and _{1}M_{0}(1; 1, n) =!n  (6)

where n! and !n are the right factorial numbers and the left factorial numbers given in (4). The function n! and !n are linked by Kurepa’s hypothesis:

KH hypothesis. For n ∈ ℕ\{1} we have

gcd( !n, n! ) = 2

where gcd(a, b) denotes the greatest common divisor of integers a and b.

This is listed as Problem B44 of Guy’s classic book [6]. In [8], it was proved that the KH is equivalent to the following assertion

!p ≢ 0 (mod p), for all primes p > 2.

The sequences a_{n}, b_{n}, c_{n}, d_{n} and e_{n} (sequences A052169, A051398, A051403, A002467 and A002720 in [17]) are defined as follows:

a_{2} = 1 a_{3} = 2 a_{n} = (n − 2)a_{n−1} + (n − 3)a_{n−2},
b_{3} = 2 b_{n} = −(n − 3)b_{n−1} + 2(n − 2)^{2},
c_{1} = 3 c_{2} = 8 c_{n} = (n + 2)(c_{n−1} − c_{n−2}),
d_{0} = 0 d_{1} = 1 d_{n} = (n − 1)(d_{n−1} + d_{n−2}),
e_{0} = 1 e_{1} = 2 e_{n} = 2ne_{n−1} − (n − 1)^{2}e_{n−2}.

They are related to the left factorial function. For instance, let p > 3 be a prime number. Then

!p ≡ −3a_{p−2} ≡ −b_{p} ≡ −2c_{p−3} ≡ d_{p−2} ≡ e_{p−1} (mod p).

We give the details for the last congruence.

Proof. Let

L_{n}^{ν}(x) = ∑_{k=0}^{n} \frac{Γ(ν + n + 1)}{Γ(ν + k + 1)} \frac{(−x)^{k}}{k!(n − k)!}

be the Laguerre polynomials, and set L_{n}^{0}(x) = L_{n}(x). Using the relation \binom{p−1}{k} ≡ (−1)^{k} (mod p), we have

L_{p−1}(x) ≡ −(p − 1)! ∑_{k=0}^{p−1} \frac{x^{k}}{k!} (mod p).

Wilson’s theorem yields

!p ≡ −L_{p−1}(−1) (mod p).

The recurrence for Laguerre polynomials

(n + 1)L_{n+1}^{ν}(x) = (ν + 2n + 1 − x)L_{n}^{ν}(x) − (ν + n)L_{n−1}^{ν}(x),

for ν = 0, x = −1 produces

h_{p−1}(p − 1)! ≡p (mod p),

where

h_{1} = 2 h_{2} = \frac{7}{2} h_{n} = 2h_{n−1} − \frac{n − 1}{n}h_{n−2}.

The identity e_{n} = h_{n}n! finally yields

!p ≡ e_{p−1} (mod p).

4 The numbers _{n}M_{−1}(1; 1, n + 1)

The special values

nM_{−1}(1; 1, n + 1) = A_{n},

are the alternating factorial numbers given in (4). This sequence satisfies the recurrence relation

A_{0} = 0, A_{1} = (−1)^{n−1}, A_{n} = −(n − 1)A_{n−1} + nA_{n−2}.

These numbers can be expressed in terms of the gamma function as follows

A_{n} = ∑_{k=1}^{n}(−1)^{n−k}Γ(k + 1) = ∫_{0}^{∞} e^{−x} (∑_{k=1}^{n}(−1)^{n−k}x^{k}) dx

= ∫_{0}^{∞} e^{−x} \frac{x^{n+1} − (−1)^{n}x}{x + 1}dx.

The same relation is now used in order to define the function A_{z}:

Definition 3 For every complex number z, Re z > 0, the function A_{z} is defined by

A_{z} =^{def} ∫_{0}^{∞} e^{−x} \frac{x^{z+1} − (−1)^{z}x}{x + 1}dx.

The identity \frac{x^{z+1}−(−1)^{z}x}{x+1} = x^{z} − \frac{x^{z}−(−1)^{z−1}x}{x+1} gives

∫_{0}^{∞} e^{−x} \frac{x^{z+1} − (−1)^{z}x}{x + 1}dx = ∫_{0}^{∞} e^{−x}x^{z}dx − ∫_{0}^{∞} e^{−x} \frac{x^{z} − (−1)^{z−1}x}{x + 1}dx,

i.e.,

A_{z} = Γ(z + 1) − A_{z−1}.  (7)

This gives A_{0} = Γ(2) − A_{1} = 0 and A_{−1} = Γ(1) − A_{0} = 1. An inductive argument shows that A_{−n}, the residue of A_{z} at the pole z = −n, is given by

res A_{−n} = (−1)^{n} ∑_{k=0}^{n−2} \frac{1}{k!}, n = 2, 3, 4, ...

The derivation employs the fact that Γ(z) is meromorphic with simple poles at z = −n and residue (−1)^{n}/n! there.

The function A_{z} can be expressed in terms of the exponential integral Ei (x) and the incomplete gamma function Γ(z, x) by

A_{z} = L[1; \frac{t^{z+1} − (−1)^{z}}{t + 1} ] = eΓ(z + 2)Γ(−z − 1, 1) − (−1)^{z}eEi (−1) − (−1)^{z}.

4.1 The generating function for A_{n−1}

The total number of arrangements of a set with n elements (sequence A000522 in [17]) is defined (see [3], [5], [15] and [16]) by:

a_{0} = 1, a_{n} = na_{n−1} + 1, or a_{n} = n! ∑_{k=0}^{n} \frac{1}{k!}.  (8)

The sequence {a_{n}} satisfies:

a_{0} = 1, a_{1} = 2, a_{n} = (n + 1)a_{n−1} − (n − 1)a_{n−2},  (9)

and

a_{0} = 1, a_{n} = ∑_{k=0}^{n−1} \binom{n}{k}(−1)^{n+1−k}(n + 1 − k)a_{k}.  (10)

Relation (9) comes from the theory of continued fractions and (10) follows directly from (8).

We now establish a connection between the sequence {a_{n}} and the alternating factorial numbers A_{n}.

Lemma 4 Let a_{−1} = 1 and n ∈ ℕ\{1}. Then

A_{n−1} = ∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}a_{k−1}.

Proof. Using (10) and induction on n we have

n! = ∑_{k=0}^{n}(−1)^{n−k}\binom{n}{k}a_{k}.

Inversion yields

a_{n} = n! − ∑_{k=1}^{n} \binom{n}{k − 1}(−1)^{n+1−k}a_{k−1}.

The relation \binom{n+1}{k} − \binom{n}{k} = \binom{n}{k−1}, k ≥ 1 produces

∑_{k=0}^{n+1} \binom{n + 1}{k}(−1)^{n+1−k}a_{k−1} + ∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}a_{k−1} = n!.

The result now follows from (7).

Theorem 3 The exponential generating function for {A_{n−1}} is given by

g(x) = e^{1−x}[E_{i}(−1) − E_{i}(x − 1) + e^{−1}] − 1 = ∑_{n=2}^{∞} A_{n−1} \frac{x^{n}}{n!} ,

where E_{i}(x) is the exponential integral (3).

Proof. The expansion of the exponential integral

E_{i}(x) = γ + ln(x) + ∑_{k=1}^{∞} \frac{x^{k}}{k · k!},

where γ is Euler’s constant, appears in [1, p. 57, 5.1.10.]. The statement of the theorem can be written as

e[E_{i}(−1) − E_{i}(x − 1) + e^{−1}] = e [− ln(x − 1) + ∑_{k=1}^{∞} \frac{(−1)^{k} − (x − 1)^{k})}{k · k!}] =

= ∑_{k=0}^{∞} a_{k−1} \frac{x^{k}}{k!} .  (11)

Expand e^{−x} as a Taylor series to obtain

e^{−x} = ∑_{k=0}^{∞}(−1)^{k} \frac{x^{k}}{k!} .

Then

e^{1−x}[E_{i}(−1) − E_{i}(x − 1) + e^{−1}] − 1 = ∑_{k=0}^{∞} a_{k−1} \frac{x^{k}}{k!} ∑_{k=0}^{∞}(−1)^{k} \frac{x^{k}}{k!} − 1

= ∑_{n=0}^{∞} ∑_{k=0}^{n} a_{k−1} \frac{x^{k}}{k!} (−1)^{n−k} \frac{x^{n−k}}{(n − k)!} − 1

= ∑_{n=0}^{∞} ∑_{k=0}^{n} \binom{n}{k}a_{k−1}(−1)^{n−k} \frac{x^{n}}{n!} − 1

= ∑_{n=0}^{∞} A_{n−1} \frac{x^{n}}{n!} − 1.

By induction on n we get

Lemma 5 Let n ∈ N. The function g(x) in Theorem 4.1 satisfies

g^{(n)}(x) = \frac{d}{dx}g^{(n−1)}(x) = (−1)^{n} [g(x) + 1 + ∑_{k=0}^{n−1} \frac{k!}{(x − 1)^{k+1}} ] .

This identity gives the algorithm for computing the n-th derivation of the function g(x).

4.2 The AL hypothesis

In [6, p. 100] the following problem is given:

Problem B43. Are there infinitely many numbers n such that A_{n−1} is a prime?

Here

A_{n} = ∑_{k=1}^{n}(−1)^{n−k}k!.

If there is a value of n − 1 such that n divides A_{n−1}, then n will divide A_{m−1} for all m > n, and there would be only a finite number of prime values. The required condition for the existence of infinitely many numbers n such that A_{n−1} is a prime may be expressed as follows:

AL hypothesis. For every prime number p

A_{p−1} ≢ 0 (mod p),

holds.

Let p be a prime number and n, m ∈ ℕ\{1}. It is not difficult to prove the following results:

A_{n−1} ≡ −1 − ∑_{k=2}^{n}[k − 1 − (−1)^{n−k}]Γ(k) (mod n),

A_{n−1} = \frac{Γ(n + 1) − 1 + ∑_{k=2}^{n−1}[(−1)^{n−k}n − k + 1 − (−1)^{n−k}]Γ(k)}{n − 1},

= 1−!n + 2 ∑_{k=1}^{n−1} A_{k}

= 3−!n−!(n − 1) · 2n + 4 ∑_{i=2}^{n} ∑_{k=1}^{i−1} A_{k}.

∑_{k=1}^{n} ∑_{i=0}^{m−1}(−1)^{i}(Γ(k + 1))^{m−i}A_{k−1}^{i} = \{ \begin{matrix}A_{n}^{m}, m even \\ A_{n}^{m} + 2 ∑_{j=1}^{n−1} A_{j}^{m}, m odd \end{matrix} ,

A_{p−1} = − ∑_{i=1}^{p} \frac{1}{Γ(i)} + p ∑_{i=1}^{p} \frac{n_{i}(−1)^{i−1}}{Γ(i)} , n_{i} ∈ ℕ (i = 1, 2, ...p)

The first step in solving problem B43 is proving the AL hypothesis. Using the previous identities, equivalents for the AL hypothesis can be given.

5 Conclusions

The main contribution is to define the function _{v}M_{m}(s; a, z) by which problems B43 and B44 are connected. The Kurepa hypothesis is an unsolved problem since 1971 and there seems to be no significant progress in solving it, apart from numerous equivalents, such as these in Section 3.3. Further details can be found in [7].

However, apart from n!, !n, and A_{n}, twenty-five more well-known sequences in [17] are special cases of the function _{v}M_{m}(s; a, z). The first study of the function gave the author the idea to find an algorithm for computing some special cases (Corollary 2 and Lemma 3) before solving the above mentioned problems.

The definition of the function _{v}M_{m}(s; a, z) suggest another method of studying the function by using the characteristics of the inverse Laplace transform.

Acknowledgements

I would like to thank the referee for the numerous comments and suggestions.

References

[1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc. New York, 1965.

[2] C. Brezinski, History of Continued Fractions and Padé Approximants, Springer-Verlag, Berlin, 1991.

[3] P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integer Sequences 3 (1) (2000), Article 00.1.5.

[4] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.

[5] J. M. Gandhi, On logarithmic numbers, Math. Student 31 (1963), 73–83.

[6] R. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994.

[7] A. Ivić and Ž. Mijajlović, On Kurepa problems in number theory, Publ. Inst. Math. (N.S.) 57 (71) (1995), 19–28.

[8] -D. Kurepa, On the left factorial function !n, Math. Balkanica 1 (1971), 147–153.

[9] -D. Kurepa, Left factorial function in complex domain, Math. Balkanica 3 (1973), 297– 307.

[10] G. V. Milovanović, A sequence of Kurepa’s functions, Sci. Rev. Ser. Sci. Eng. No. 19–20 (1996), 137–146.

[11] G. V. Milovanović and A. Petojević, Generalized factorial function, numbers and polynomials and related problems, Math. Balkanica, to appear.

[12] O. Perron, Die Lehren von den Kettenbüchen, Chelsea Publishing Company, New York, 1954.

[13] A. Petojević, On Kurepa’s hypothesis for left factorial, Filomat (Nis), 12 (1) (1998), 29–37.

[14] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions, Nauka, Moscow, 1981. (Russian)

[15] J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958.

[16] D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student 20 (1952), 66–70.

[17] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at

    http://www.research.att.com/~njas/sequences/

[18] H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990.

2000 Mathematics Subject Classification: Primary 11B83; Secondary 33C05, 44A10. Keywords: left factorial, alternating factorial, hypergeometric function, Laplace transform

(Concerned with sequences \underline{A000142}, \underline{A003422}, \underline{A005165}, \underline{A000217}, \underline{A014144}, \underline{A007489}, \underline{A000292}, \underline{A000332}, \underline{A000389}, \underline{A014145}, \underline{A000166}, \underline{A000125}, \underline{A000217}, \underline{A000389}, \underline{A055795}, \underline{A027660}, \underline{A055796}, \underline{A055797}, \underline{A032179}, \underline{A032031}, \underline{A001710}, \underline{A001715}, \underline{A001720}, \underline{A001725}, \underline{A066318}, \underline{A000165}, \underline{A047053}, \underline{A014297}, \underline{A052169}, \underline{A051398}, \underline{A051403}, \underline{A002467}, \underline{A002720}, and \underline{A000522}.)

Received March 21, 2002; revised version received August 14, 2002. Published in Journal of Integer Sequences August 31, 2002.

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