Table Of ContentNew Congruences and Finite Difference
Equations for Generalized Factorial Functions
Maxie D. Schmidt
University of Washington
Department of Mathematics
7
1 Padelford Hall
0
Seattle, WA 98195
2
n USA
a
J [email protected]
7
1
] Abstract
O
C We use the rationality of the generalized hth convergent functions, Convh(α,R;z),
. to the infinite J-fraction expansions enumerating the generalized factorial product se-
h
quences, p (α,R) = R(R+α)···(R+(n−1)α), defined in the references to construct
t n
a
new congruences and h-order finite difference equations for generalized factorial func-
m
tions modulo hαt for any primes or odd integers h ≥ 2 and integers 0≤ t ≤ h. Special
[
cases of the results we consider within the article include applications to new congru-
1
ences and exact formulas for the α-factorial functions, n! . Applications of the new
v (α)
1 results we consider within the article include new finite sums for the α-factorial func-
4 tions, restatements of classical necessary and sufficient conditions of the primality of
7
special integer subsequences and tuples, and new finite sums for the single and double
4
0 factorial functions modulo integers h ≥ 2.
.
1
0
7
1 1 Notation and other conventions in the article
:
v
i
X 1.1 Notation and special sequences
r
a
Most of the conventions in the article are consistent with the notation employed within the
Concrete Mathematics reference, and the conventions defined in the introduction to the first
articles [11, 12]. These conventions include the following particular notational variants:
◮ Extraction of formal power series coefficients. The special notation for formal
power series coefficient extraction, [zn] f zk :7→ f ;
k k n
◮ Iverson’s convention. The more com(cid:0)Ppact usa(cid:1)ge of Iverson’s convention, [i = j] ≡
δ
δ , in place of Kronecker’s delta function where [n = k = 0] ≡ δ δ ;
i,j δ n,0 k,0
1
◮ Bracket notation for the Stirling number triangles. We use the alternate
bracket notation for the Stirling number triangles, n = (−1)n−ks(n,k) and n =
k k
S(n,k);
(cid:2) (cid:3) (cid:8) (cid:9)
◮ Harmonic number sequences. Use of the notation for the first-order harmonic
(1)
numbers, H or H , which defines the sequence
n n
1 1 1
H := 1+ + +···+ ,
n
2 3 n
andthenotationforthepartial sums forthemoregeneral cases ofther-orderharmonic
(r)
numbers, H , defined as
n
H(r) := 1+2−r +3−r +···+n−r,
n
(r)
when r,n ≥ 1 are integer-valued and where H ≡ 0 for all n ≤ 0;
n
◮ Rising and falling factorial functions. We use the convention of denoting the
falling factorial function by xn = x!/(x − n)!, the rising factorial function as xn =
Γ(x + n)/Γ(x), or equivalently by the Pochhammer symbol, (x) = x(x + 1)(x +
n
2)···(x+n−1);
◮ Shorthand notation in integer congruences and modular arithmetic. Within
the article the notation g (n) ≡ g (n) (mod N ,N ,...,N ) is understood to mean
1 2 1 2 k
that the congruence, g (n) ≡ g (n) (mod N ), holds modulo any of the bases, N , for
1 2 j j
1 ≤ j ≤ k.
The standard set notation for Z, Q, and R denote the sets of integers, rational numbers,
and real numbers, respectively, where the set of natural numbers, N, is defined by N :=
{0,1,2,...} = Z+ {0}. Other more standard notation for the special functions cited within
the article is consistent with the definitions employed inthe NIST Handbook of Mathematical
S
Functions (2010).
1.2 Mathematica summary notebook document and computational
reference information
Thearticleispreparedwithamoreextensive set ofcomputationaldataandsoftwareroutines
released as open source software to accompany the examples and numerous other applica-
tions suggested as topics for future research and investigation within the article. It is highly
encouraged, and expected, that theinterested reader obtaina copy ofthe summary notebook
reference and computational documentation prepared in this format to assist with compu-
tations in a multitude of special case examples cited as particular applications of the new
results.
The prepared summary notebook file, multifact-cfracs-summary.nb , attached to the
submission of this manuscript contains the working Mathematica code to verify the formulas,
2
propositions, andotheridentitiescitedwithinthearticle[13]. Giventhelengthofthisandthe
first article, the Mathematica summary notebook included with this submission is intended
to help the reader with verifying and modifying the examples presented asapplications of the
new results cited below. The summary notebook also contains numerical data corresponding
tocomputationsofmultipleexamplesandcongruencesspecificallyreferencedinseveralplaces
by the applications given in the next sections of the article.
2 Introduction
2.1 Motivation
In this article, we extend the results from the reference [11] providing infinite J-fraction
expansions for the typically divergent ordinary generating functions (OGFs) of generalized
factorial product sequences of the form
p (α,R) := R(R+α)(R+2α)×···×(R+(n−1)α)[n ≥ 1] +[n = 0] , (1)
n δ δ
whenR depends linearlyonn. Notablespecial cases of (1)thatweareparticularlyinterested
in enumerating through the convergents to these J-fraction expansions include the multiple,
or α-factorial functions, n! , defined for α ∈ Z+ as
(α)
n·(n−α)! , if n > 0;
(α)
n! = 1, if −α < n ≤ 0; (2)
(α)
0, otherwise,
and the generalized factorial functions of the form p (α,βn + γ) for α,β,γ ∈ Z, α 6= 0,
n
and β,γ not both zero. The second class of special case products are related to the Gould
polynomials, G (x;a,b) = x · x−an n, through the following identity ([12, §3.4.2],[10,
n x−an b
§4.1.4]):
(cid:0) (cid:1)
(−α)n+1
p (α,βn+γ) = ×G (γ −α−β;−β,−α). (3)
n n+1
γ −α−β
The α-factorial functions, (αn−d)! for α ∈ Z+ and some 0 ≤ d < α, form special cases
(α)
of (3) where, equivalently, (α,β,γ) ≡ (−α,α,−d) and (α,β,γ) ≡ (α,0,α−d) [11, §6]. The
α-factorial functions are expanded by the triangles of Stirling numbers of the first kind, n ,
k
and the α-factorial coefficients, n , respectively, in the following forms [5, 12]:
k α (cid:2) (cid:3)
n (cid:2) (cid:3)
⌈n/α⌉
n!(α) = (−α)⌈αn⌉−mnm, ∀n ≥ 1,α ∈ Z+ (4)
m
m=0(cid:20) (cid:21)
X
= n ⌊n−α1+α⌋+1 (−1)⌊n−α1+α⌋−m(n+1)m, ∀n ≥ 1,α ∈ Z+
m+1
m=0(cid:20) (cid:21)α
X
3
n
n
(αn−d)! = (α−d)× (−1)n−m(αn+1−d)m−1
(α)
m
m=1(cid:20) (cid:21)α
X
n
n+1
= (−1)n−m(αn+1−d)m, ∀n ≥ 1,α ∈ Z+,0 ≤ d < α.
m+1
m=0(cid:20) (cid:21)α
X
A careful treatment of the polynomial expansions of these generalized α-factorial functions
through the coefficient triangles in (4) is given in the reference [12].
2.2 Summary of the J-fraction results
For all h ≥ 2, we can generate the generalized factorial product sequences, p (α,R), through
n
the strictly rational generating functions provided by the hth convergent functions, denoted
by Conv (α,R;z), to the infinite continued fraction series established by the reference [11].
h
In particular, we have series expansions of these convergent functions given by
1
Conv (α,R;z) :=
h αR·z2
1−R·z −
2α(R+α)·z2
1−(R+2α)·z −
3α(R+2α)·z2
1−(R+4α)·z −
···
1−(R+2(h−1)α)·z
FP (α,R;z)
h
= (5)
FQ (α,R;z)
h
h ∞
= p (α,R)zn + [p (α,R) (mod h)]zn,
n n
n=0 n>h
X X
where the convergent function numerator and denominator polynomial subsequences provid-
ing the characteristic expansions of (5) are given in closed-form by
h k−1
h
FQ (α,R;z) = (−1)k (R+(h−1−j)α) zk (6)
h k
!
k=0(cid:18) (cid:19) j=0
X Y
h
h R
= +h−k (−αz)k
k α
k=0(cid:18) (cid:19)(cid:18) (cid:19)k
X
= (−αz)h ·h!×L(R/α−1) (αz)−1 ,
h
(β) (cid:0) (cid:1)
when L (x) denotes an associated Laguerre polynomial, and where
n
h−1
FP (α,R;z) = C (α,R)zn (7a)
h h,n
n=0
X
4
h−1 n
h
= (−1)ip (−α,R+(h−1)α)p (α,R) zn (7b)
i n−i
i
!
n=0 i=0 (cid:18) (cid:19)
X X
h−1 n
h
= (1−h−R/α) (R/α) (αz)n. (7c)
i i n−i
!
n=0 i=0 (cid:18) (cid:19)
X X
Thecoefficientsofthepolynomialpowersofz inthepreviousseveralexpansions, C (α,R) :=
h,n
[zn]FP (α,R;z) for 0 ≤ n < h, also have the following multiple, alternating sum expansions
h
involving the Stirling number triangles [11, §5.2]:
m−s
h m k R R
C (α,R) = (−1)mαn −1 ×hs (8a)
h,n
k s m α α
0≤m≤k≤n (cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:21) (cid:18) (cid:19)n−k(cid:18) (cid:19) !
X
0≤s≤n
h m k n−k
= (−1)mαn−s(h−1)m−t ×Rs (8b)
k t m s−t
0≤m≤k≤n(cid:18)(cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:21)(cid:20) (cid:21) (cid:19)
X
0≤t≤s≤n
m−s
h h m k s R R
= (−1)mαn −1 ×i! (8c)
k i s m i α α
0≤m≤k≤n(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:21)(cid:26) (cid:27) (cid:18) (cid:19)n−k(cid:18) (cid:19)
X
0≤i≤s≤n
h m i h+v k s
= (−1)m+i−vαn× (8d)
k s v v m i
0≤m≤k≤n (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:21)(cid:26) (cid:27)
X
0≤v≤i≤s≤n
m−s
R R
× −1 ×i!
α α
(cid:18) (cid:19)n−k(cid:18) (cid:19)
n
h m k n−k s R
= (−1)m+s−i(h−1)m−t ×αn .
k t m s−t i α
i=0 0≤m≤k≤n(cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:21)(cid:20) (cid:21)(cid:26) (cid:27) (cid:18) (cid:19)i
X X
0≤t≤s≤n
polynomial function of h only
(8e)
| {z }
Given that the non-zero hth convergent functions, Conv (α,R;z), are rational in each of
h
z,α,R for all h ≥ 1, and that the convergent denominator sequences, FQ (α,R;z), have
h
characteristic expansions by the Laguerre polynomials and the confluent hypergeometric
functions, we may expand both exact finite sums and congruences modulo hαt for the gen-
eralized factorial functions, p (α,βn + γ), by the distinct special zeros of these functions
n
in the next forms when the hth convergent functions are expanded in partial fractions as
Conv (α,R;z) ≡ c (α,R)/(1−ℓ (α,R)·z) [11, §6.2].
h 1≤j≤h h,j h,j
Pn
p (α,R) = c (α,R)×ℓ (α,R)n (9)
n n,j n,j
j=1
X
5
h
p (α,R) ≡ c (α,R)×ℓ (α,R)n (mod h)
n h,j h,j
j=1
X
n
n!(α) = cn,j(−α,n)×ℓn,j(−α,n)⌊n−α1⌋
j=1
X
h
n!(α) ≡ ch,j(−α,n)×ℓh,j(−α,n)⌊nα−1⌋ (mod h,hα,··· ,hαh)
j=1
X
2.3 Key new results proved in the article
Whereas in the first reference [11] we prove new exact formulas and congruence properties
using the algebraic properties of the rational convergents, Conv (α,R;z), in this article we
h
choose an alternate route to derive our new results. Namely, we use the rationality of the hth
convergent functions to establish new h-order finite difference equations for the coefficients
of Conv (α,βn+γ;z), both in the exact forms of p (α,βn+γ) with respect to n, as well
h n
as for these special factorial functions expansions modulo hαt for any integers h ≥ 2 and
0 ≤ t ≤ h. We state the next key proposition, which we subsequently prove in Section 3.1,
before giving several examples of the applications we consider within the article.
Proposition 1 (Finite Difference Equations for Generalized Factorial Functions). For fixed
α,β,γ ∈ Z with α 6= 0 and β,γ not both zero, h odd or prime, an integer 0 ≤ t ≤ h, and
any integers n,r ≥ 0, we have the next exact expansions and congruences for the generalized
product sequences, p (α,R) and p (α,βn+γ), where the numerator coefficients, C (α,R),
n n h,n
are given by C (α,R) := [zk]FP (α,R;z).
h,k h
n−1
p (α,R) = n+r (−1)kp (−α,R+(n−1+r)α)p (α,R)+C (α,R)
n k+1 k+1 n−1−k n+r,n
k=0
X(cid:0) (cid:1)
(10a)
n−1
= n (−1)kp (−α,R+(n−1)α)p (α,R)+[n = 0]
k+1 k+1 n−1−k δ
k=0
X(cid:0) (cid:1)
n
h
p (α,βn+γ) ≡ (−α)kp (−α,βn+γ +(h−1)α)p (α,βn+γ) (mod h)
n k n−k
k
k=0(cid:18) (cid:19)
X
(10b)
n
h
= αn+k 1−h− βn+γ βn+γ
k α k α n−k
k=0(cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
n
h
p (α,βn+γ) ≡ αn+(t+1)k 1−h− βn+γ βn+γ (mod hαt) (10c)
n k α k α n−k
k=0(cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
6
Remark 2 (Stronger Statements of the Key Congruence Properties). Based on numerical
evidence computed in the summary notebook reference [13], we conjecture, but do not offer
conclusive proof here, that the results stated in (10b) and (10c) of the key proposition in
fact hold for all integers h ≥ 2, α 6= 0, and 0 ≤ t ≤ h – not just in the odd and even
prime special cases of the moduli specified in the previous proposition. This observation
substantially widens the utility of the application of these results in the examples cited in
Section 3 below. Morevover, the observation of this compuationally-verified result should be
considered in evaluating the significance of the implications of these applications in the new
contexts suggested by Section 3.6 as generalized forms of our approach.
2.4 Examples
2.4.1 Applications in Wilson’s theorem and Clement’s theorem concerning the
twin primes
The first examples given in this section provide restatements of the necessary and sufficient
integer-congruence-based conditions imposed in both statements of Wilson’s theorem and
Clement’s theorem through the exact expansions of the factorial functions defined above.
For odd integers p ≥ 3, the congruences implicit to each of Wilson’s theorem and Clement’s
theorem are enumerated as follows [9, §4.3] [6, §6.6] [2]:
p prime ⇐⇒ (p−1)!+1 ≡ 0 (mod p)
⇐⇒ [zp−1]Conv (−1,p−1;z)+1 ≡ 0 (mod p)
p
⇐⇒ [zp−1]Conv (1,1;z)+1 ≡ 0 (mod p)
p
p,p+2 prime ⇐⇒ 4((p−1)!+1)+p ≡ 0 (mod p(p+2))
⇐⇒ 4[zp−1]Conv (−1,p−1;z)+p+4 ≡ 0 (mod p(p+2))
p(p+2)
⇐⇒ 4[zp−1]Conv (1,1;z)+p+4 ≡ 0 (mod p(p+2)).
p(p+2)
The rationality in z of the convergent functions, Conv (α,R;z), at each h leads to further
h
alternate formulations of other well-known congruence statements concerning the divisibility
of factorial functions. For example, we may characterize the primality of the odd integers,
p > 3, of the form p = 4k+1 (i.e., the so-termed subset of “Pythagorean primes”) according
to the next condition [6, §7] (A002144).
p−1 !2 ≡ −1 (mod p) ⇐⇒ p is a prime of the form 4k +1 (11)
2
For an odd in(cid:0)teger(cid:1)p > 3 to be both prime and satisfy p ≡ 1 (mod 4), the congruence
statement in(11)requires thatthediagonalsofthefollowingrationaltwo-variableconvergent
generating functions satisfy the following equivalent conditions where p is chosen so that
i
p | 2pp for each i = 1,2:
i
p−1 p−1 z
p−1 !2 = [z(p−1)/2][x0] Conv −1, ;x Conv −1, ; ≡ −1 (mod p)
2 p1 2 p2 2 x
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(cid:0) (cid:1)
7
z
p−1 !2 = [z(p−1)/2][x0] Conv (−2,p−1;x)Conv −2,p−1; ≡ −1 (mod p)
2 p1 p2 4x
(cid:0) (cid:1) (cid:16) p−1 (cid:16) (cid:17)z(cid:17)
p−1 !2 = [z(p−1)/2][x0] Conv −1, ;x Conv −2,p−1; ≡ −1 (mod p).
2 p1 2 p2 2x
(cid:18) (cid:18) (cid:19) (cid:19)
(cid:16) (cid:17)
(cid:0) (cid:1)
The reference provides remarks on the harmonic-number-related fractional power series ex-
pansions of the convergent-based generating functions, Conv (1,1;z/x)×Conv (1,1;x) and
n n
Conv (2,1;z/x)×Conv (1,1;x), related to the single factorial function squares enumerated
n n
by the identities in the previous equations [13].
These particular congruences involving the expansions of the single factorial function are
considered in the reference [12, §6.1.6] as an example of the first product-based symbolic
factorial function expansions implicit to both Wilson’s theorem and Clement’s theorem.
Related formulations of conditions concerning the primality of prime pairs, (p,p + d), and
then of other prime k-tuples, are similarly straightforward to obtain by elementary methods
starting from the statement of Wilson’s theorem [8]. For example, the new results proved in
Section 3 are combined with the known congruences established in the reference [8, §3, §5] to
obtain the cases of the next particular forms of alternate necessary and sufficient conditions
for the twin primality of the odd positive integers p := 2n+1 and p := 2n+3 when n ≥ 1
1 2
(A001359, A001097):
2n+1,2n+3 odd primes (12)
n 2
⇐⇒ 2 (2n+1)(2n+3) 2(−1)ii!(n−i)! +(−1)n(10n+7) ≡ 0
i
(cid:18)i=0 (cid:19)
P(cid:0) (cid:1)
(mod (2n+1)(2n+3))
2n
⇐⇒ 4 (2n+1)(2n+3) 2(−1)ii!(2n−i)! +2n+5 ≡ 0
i
(cid:18)i=0 (cid:19)
P(cid:0) (cid:1)
(mod (2n+1)(2n+3)).
Section 3.4.2 of this article considers the particular cases of these two classically-phrased
congruence statements as applications of the new polynomial expansions for the generalized
product sequences, p (α,βn+γ), derived from the expansions of the convergent function
n
sequences by finite difference equations.
2.4.2 Congruences for the Wilson primes and the single factorial function mod-
ulo n2
The sequence of Wilson primes, or the subsequence of odd integers p ≥ 5 satisfying n2 |
(n−1)!+1, is characterized through each of the following additional divisibility requirements
placed on the expansions of the single factorial function implicit to Wilson’s theorem cited
8
by the applications of the new results given below in Section 3.4 of the article (A007540):
n−1 n2 2
(−1)ii!(n−1−i)! ≡ −1 (mod n2)
i
i=0 (cid:18) (cid:19)
X
≡ (n−1)! (mod n2)
n|−1 n2 {z }
(n2 −n)i ×(−1)n−1−i(n−1)n−1−i ≡ −1 (mod n2)
i
i=0 (cid:18) (cid:19)
X
≡ (n−1)! (mod n2)
n−1 s i n−1 k | {z }
n2 m i n2+v k s (−1)i−v(n−1)n−1−k(−n)m−si! ≡ −1. (mod n2)
k s v v m i
s=0i=0v=0(cid:18)k=0m=0 (cid:19)
P P P P P (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1)(cid:2) (cid:3)(cid:8) (cid:9)
The results providing the new congruence properties for the α-factorial functions modulo
the integers p, and pαi for some 0 ≤ i ≤ p, expanded in Section 3.2 also lead to alter-
nate phrasings of the necessary and sufficient conditions on the primality of several notable
subsequences of the odd positive integers n ≥ 3 [11, cf. §6.4].
2.5 Expansions of congruences for the α-factorial functions and
related sequences
2.5.1 Expansions of other new congruences for the double and triple factorial
functions
A few representative examples of the new congruences for the double and triple factorial
functions obtained from the statements of Corollary 4 given in Section 3 also include the
following particular expansions for integers p ,p ≥ 2, and where 0 ≤ s ≤ p and 0 ≤
1 2 1
t ≤ p assume some prescribed values over the non-negative integers (see the computations
2
contained in the reference [13]):
n
p
(2n−1)!! ≡ 1 2n(−2)(s+1)i 1 −p 1 (mod p 2s)
i 2 1 i 2 n−i 1
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
n
p
≡ 1 (−2)n2(s+1)i 1 +n−p 1 −n+i (mod p 2s)
i 2 1 i 2 n−i 1
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
n
p
(3n−1)!!! ≡ 2 3n(−3)(t+1)i 1 −p 2 (mod p 3t)
i 3 2 i 3 n−i 2
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
n
p
≡ 2 (−3)n3(t+1)i 1 +n−p 1 −n+i (mod p 3t)
i 3 2 i 3 n−i 2
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
n
p
(3n−2)!!! ≡ 2 3n(−3)(t+1)i 2 −p 1 (mod p 3t)
i 3 2 i 3 n−i 2
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
9
n
p
≡ 2 (−3)n3(t+1)i 2 +n−p 2 −n+i (mod p 3t).
i 3 2 i 3 n−i 2
i=0 (cid:18) (cid:19)
X (cid:0) (cid:1) (cid:0) (cid:1)
2.5.2 Semi-polynomial congruences for double factorial functions and the cen-
tral binomial coefficients
The integer congruences satisfied by the double factorial function, (2n − 1)!!, and the
Pochhammer symbol cases, 2n × 1 , expanded in Section 3.5 provide the next variants
2 n
of the polynomial congruences for the central binomial coefficients, 2n = 2n×(2n−1)!!/n!,
(cid:0) (cid:1) n
reduced modulo the respective integer multiples of 2n + 1 and the polynomial powers, np,
(cid:0) (cid:1)
for fixed integers p ≥ 2 in the following equations (A000984) (see the computations in the
reference [13])1.
n
2n
≡ 2x+1 (−2)i(1 +2x)i(1) × 22n (mod 2n+1)
n i 2 2 n−i n!
(cid:18) (cid:19) Xi=0 (cid:0) m(cid:1)od 2x+1 # x :7→ n
2n
≡ n xp 2n−2i (1 −xp) × 8i·(n−i)! (mod np)
n i=0 i n−i 2 i n!
(cid:18) (cid:19) mod xp # x :7→ n
P (cid:0) (cid:1)(cid:0) (cid:1)
3 Finite difference equations for generalized factorial
functions and applications
The rationality of the convergent functions, Conv (α,R;z), in z for all h suggests new
h
forms of h-order finite difference equations with respect to h, α, and R satisfied by the
product sequences, p (α,R), when α and R correspond to fixed parameters independent
n
of the sequence indices n. In particular, the rationality of the hth convergent functions
immediately implies the first two results stated in Proposition 1 above, which also provides
both forms of the congruence properties stated in (10b) modulo odd and even prime integers
h ≥ 2 [7, §2.3] [5, §7.2].
Whentheinitiallyindeterminateparameter, R,assumesanimplicitdependenceonthese-
quence index, n, theresults phrasedbytheprevious equations, somewhat counter-intuitively,
do not immediately imply difference equations satisfied between only thegeneralized product
sequences, either exactly, or modulo the prescribed choices of h ≥ 2 (cf. (13) and Example 3
onpage12). The newformulas connecting thegeneralized product sequences, p (α,βn+γ),
n
resulting from (10b) and (10c) in these cases are, however, reminiscent of the relations sat-
isfied between the generalized Stirling polynomial and convolution polynomial sequences
expanded in the references [12] [5, cf. §6.2].
1 The unconventional notation for computing the underlined polynomial mod operations in the next
equations is defined by the footnote on page 31.
10
