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Factorial polynomials

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The factorial polynomials are to [finite] difference calculus what polynomials are to [infinitesimal] differential calculus.

Falling factorial and raising factorial

The falling factorial is defined as[1]

and the raising factorial is defined as[1]

where, in either case, for
k = 0
we get the empty product, i.e.
1
.

Factorial polynomials

A factorial term (Boole, 1970: p. 6) or a factorial polynomial (Elaydi, 2005: p. 60) is defined as

and for negative
k
we have the definition
where, in either case, for
k = 0
we get the empty product, i.e.
1
.
The factorial polynomials of degree
n
are defined as the sum of factorial terms
where for
n = 0
we get the constant polynomial
a0
.

In a fashion similar to Laurent polynomials, we also have the definition

Finite difference operator

With the above definitions, the [finite] difference operator

where
E
is the shift operator and
I
is the identity operator, behaves with the [whether ordinary or generalized] factorial polynomials like the [infinitesimal] differential operator does with the [whether ordinary or Laurent] polynomials.

Difference operator Differential operator

Stirling numbers

For
n   ≥   0
, we have

and

where
s(n, k)
or
S1(n, k)
are Stirling numbers of the first kind, and are unsigned Stirling numbers of the first kind.[2]

For
n   ≥   0
, we have

and

where ,
S (n)k
,
S(n, k)
or
S2(n, k)
are Stirling numbers of the second kind, and
( − 1)n+k S2(n, k)
are signed Stirling numbers of the second kind.[2]

Stirling numbers of the first kind

We have

involving the triangle of Stirling numbers of the first kind, where the coefficients of row
k
are obtained by multiplying the polynomial of row
k  −  1
by
(x  −  k + 1)
.

A048994 Triangle read by rows of Stirling numbers of the first kind,
S1(n, k), n   ≥   0, 0   ≤   k   ≤   n
.
{1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -6, 11, -6, 1, 0, 24, -50, 35, -10, 1, 0, -120, 274, -225, 85, -15, 1, 0, 720, -1764, 1624, -735, 175, -21, 1, 0, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, ...}
A008275 Triangle read by rows of Stirling numbers of the first kind,
S1(n, k), n   ≥   1, 1   ≤   k   ≤   n
.
{1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50, 35, -10, 1, -120, 274, -225, 85, -15, 1, 720, -1764, 1624, -735, 175, -21, 1, -5040, 13068, -13132, 6769, -1960, 322, -28, 1, ...}

Row sums of Stirling numbers of the first kind

The row sums of Stirling numbers of the first kind are

Unsigned Stirling numbers of the first kind

We have

involving the triangle of unsigned Stirling numbers of the first kind, also called the factorial triangle,[3] where the coefficients of row
k
are obtained by multiplying the polynomial of row
k  −  1
by
(x + k  −  1)
.

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows,
| S1(n, k) |
, n   ≥   0, 0   ≤   k   ≤   n
.
{1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, ...}

Row sums of triangle of unsigned Stirling numbers of the first kind

The row sums of unsigned Stirling numbers of the first kind are

Falling diagonals of triangle of unsigned Stirling numbers of the first kind

The coefficients of the
k
th falling diagonal, where
k = 0
refers to the rightmost one, are given by a polynomial of degree
2k
.

A000012 Stirling numbers of the first kind:
s(n, n), n   ≥   0.
(The simplest sequence of positive numbers: the all 1's sequence.)

[a_0(n) = 1] G.f.: 1/(1-x).

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}
A000217 Stirling numbers of the first kind:
s(n + 1, n), n   ≥   0.
(Triangular numbers:
a(n) = C(n + 1, 2) = n (n + 1) / 2 = 0 + 1 + 2 + ... + n
.)

[a_1(n) = (n + n^2) / 2 = n (1 + n) / 2] G.f.: x / (1-x)^3.

{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, ...}
A000914 Stirling numbers of the first kind:
s(n + 2, n), n   ≥   0
.

[a_2(n) = (10 n + 21 n^2 + 14 n^3 + 3 n^4) / 24 = n (1 + n) (2 + n) (5 + 3 n) / 24] G.f.: (2*x + x^2) / (1-x)^5.

{0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, ...}
A001303 Stirling numbers of first kind,
s(n + 3, n), n   ≥   1,
negated.

[a_3(n) = (36 n + 96 n^2 + 97 n^3 + 47 n^4 + 11 n^5 + n^6) / 48 = n (1 + n) (2 + n)^2 (3 + n)^2 / 48] G.f.: (6*x + 8*x^2 + x^3) / (1-x)^7.

{6, 50, 225, 735, 1960, 4536, 9450, 18150, 32670, 55770, 91091, 143325, 218400, 323680, 468180, 662796, 920550, 1256850, 1689765, 2240315, 2932776, 3795000, 4858750, 6160050, ...}
A000915 Stirling numbers of first kind
s(n + 4, n), n   ≥   1
.

[a_4(n) = ?] G.f.: (24*x + 58*x^2 + 22*x^3 + x^4) / (1-x)^9.

{24, 274, 1624, 6769, 22449, 63273, 157773, 357423, 749463, 1474473, 2749747, 4899622, 8394022, 13896582, 22323822, 34916946, 53327946, 79721796, 116896626, 16842387, ...}

Triangle of polynomials for the numerator of the o.g.f.s for unsigned Stirling numbers of first kind along the falling diagonals:



A112007 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals, for
n   ≥   1, 1   ≤   k   ≤   n
.
{1, 2, 1, 6, 8, 1, 24, 58, 22, 1, 120, 444, 328, 52, 1, 720, 3708, 4400, 1452, 114, 1, 5040, 33984, 58140, 32120, 5610, 240, 1, 40320, 341136, 785304, 644020, 195800, 19950, ...}

Stirling numbers of the second kind

We have

involving the triangle of Stirling numbers of the second kind.

A048993 Triangle of Stirling numbers of second kind,
S2(n, k), n   ≥   0, 0   ≤   k   ≤   n
.
{1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, ...}
A106800 Triangle of Stirling numbers of second kind,
S2(n, n  −  k), n   ≥   0, 0   ≤   k   ≤   n
.
{1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 1, 0, 1, 21, 140, 350, 301, 63, 1, 0, 1, 28, 266, 1050, 1701, 966, 127, 1, 0, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, ...}
A008277 Triangle read by rows of Stirling numbers of the second kind,
S2(n, k), n   ≥   1, 1   ≤   k   ≤   n
.
{1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1, 1, 31, 90, 65, 15, 1, 1, 63, 301, 350, 140, 21, 1, 1, 127, 966, 1701, 1050, 266, 28, 1, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, ...}

Row sums of triangle of Stirling numbers of the second kind

The row sums of Stirling numbers of the second kind are

where
Bn
are the Bell (or exponential) numbers.

A000110 Bell or exponential numbers: number of ways to partition a set of
n, n   ≥   0,
labeled elements.
{1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, ...}

Bell polynomials

By replacing the
x (k)
by
xk
in

one obtains the Bell polynomials

for which
Bn (1)
yields the Bell numbers and
Bn ( − 1)
yields the complementary Bell numbers.

Signed Stirling numbers of the second kind

We have

involving the triangle of signed Stirling numbers of the second kind.

A?????? Triangle of signed Stirling numbers of second kind,
( − 1)n+k S2(n, k), n   ≥   0, 0   ≤   k   ≤   n
.
{1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, -1, 7, -6, 1, 0, 1, -15, 25, -10, 1, 0, -1, 31, -90, 65, -15, 1, 0, 1, -63, 301, -350, 140, -21, 1, 0, -1, 127, -966, 1701, -1050, 266, -28, 1, ...}
A?????? Triangle read by rows of signed Stirling numbers of the second kind,
( − 1)n+k S2(n, k), n   ≥   1, 1   ≤   k   ≤   n
.
{, ...}

Row sums of triangle of signed Stirling numbers of the second kind

The row sums of signed Stirling numbers of the second kind

where
n
are the Rao Uppuluri-Carpenter numbers (or complementary Bell numbers), yield the sequence
{1, (-) -1, 0, (-) 1, 1, (-) -2, -9, (-) -9, 50, (-) 267, 413, (-) -2180, -17731, (-) -50533, 110176, (-) 1966797, 9938669, (-) 8638718, -278475061, (-) -2540956509, -9816860358, (-) 27172288399, ...}


A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers),
n   ≥   0
: e.g.f. =
exp(1  −  exp(x))
.
{1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731, -50533, 110176, 1966797, 9938669, 8638718, -278475061, -2540956509, -9816860358, 27172288399, 725503033401, 5592543175252, ...}

Factorial polynomials inspired from ordinary binomial expansions

Factorial polynomials inspired from ordinary binomial expansion of (x+1)^n

By replacing the
xk
by
x (k)
in the binomial expansion

we get

Examples:

Triangle read by rows of ordinary polynomial coefficients resulting from replacing x^k by x^(k) in the binomial expansion of (x+1)^n, for
n   ≥   0
.
{1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 5, -2, 1, 1, 9, -15, 15, -5, 1, 1, -35, 94, -85, 40, -9, 1, ...}

Triangle row sums

It appears that the row sums yield

Triangle columns

It appears that we might have for column 1 =
{ 0, 1, 1, 2, 0, 9, -35, ...}
where
[ · ]
is the Iverson bracket.

A002741 Logarithmic numbers: expansion of
 −  log(1  −  x) exp( −  x)
.
{0, 1, -1, 2, 0, 9, 35, 230, 1624, 13209, 120287, 1214674, 13469896, 162744945, 2128047987, 29943053062, 451123462672, 7245940789073, ...}

Factorial polynomials inspired from ordinary binomial expansion of (x-1)^n

By replacing the
xk
by
x (k)
in the binomial expansion

we get

Examples:

A?????? Triangle read by rows of ordinary polynomial coefficients resulting from replacing x^k by x^(k) in the binomial expansion of (x-1)^n, for
n   ≥   0
.
{1, -1, 1, 1, -3, 1, -1, 8, -6, 1, 1, -24, 29, -10, 1, ...}

Related to (although the signs differ)

A094816 Triangle read by rows: T(n,k), 0<=k<=n, = coefficients of Charlier polynomials: A046716 transposed.

{1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 24, 29, 10, 1, 1, 89, 145, 75, 15, 1, 1, 415, 814, 545, 160, 21, 1, 1, 2372, 5243, 4179, 1575, 301, ...}

Triangle row sums

It appears that the row sums yield

Triangle columns

It appears that we have for column 1 =
{ 0, 1, -3, 8, -24, 89, -415, ...}
A002104 Logarithmic numbers: expansion of
 −  log(1  −  x) exp(x)
.
{0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, ...}

Notes

  1. 1.0 1.1 Using the
    xk
    and
    xk
    notation proposed in (Graham, Knuth, and Patashnik, 1994: p. 48).
  2. 2.0 2.1 Named after the Scottish mathematician James Stirling.
  3. Steven Schwartzman, "The Factorial Triangle and Polynomial Sequences", The College Mathematics Journal, Vol. 15, No. 5 (Nov., 1984), pp. 424-426.

References

  • Boole, G. Finite Differences, 5th ed. New York, NY: Chelsea, 1970.
  • Elaydi, S.N. An Introduction to Difference Equations, 3rd ed. Springer, 2005.
  • Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

External links