1. Introduction
Peng, J. has introduced Shape of numbers in [1] [2] [3]:
,
,
. There are M − 1 intervals between adjacent numbers.
means continuity,
means discontinuity.
Shape of numbers: collect
with the same continuity and discontinuity at the same position into a catalog, call it a Shape.
A shape has a min Item:
that use the symbol PS = [min Item] to represent it.
If
, only
is allowed. If
, only
is allowed.
The single
is an item,
is the product. Ii is a factor.
Example:
,
,
Define:
SET(N, PS) = set of items belonging to PS in [1, N − 1]
PM(PS) = count of factors
PB(PS) = count of discontinuities
MIN(PS) = min product:
,
IDX(PS) = (max factor) + 1
PH(PS) = IDX(PS) − PB(PS) − 2
Basic Shape: intervals = 1 or 2
BASE(PS) = BS: if (1) PB(BS) = PB(PS), (2) PM(BS) = PM(PS), (3) BS is a Basic Shape, (4) BS has discontinuity intervals at the same positions of PS.
Example:
End(N, PS) = set of items belonging to PS with the max factor = N − 1;
|SET(N, PS)| = count of items in SET(N, PS);
SUM(N, PS) = sum of all products in SET(N, PS).
Example:
[3] introduced the subset:
If PB(PS) = 0, SET(N, PS) is simple.
If PB(PS) > 0, then can fix some interval of discontinuities to get subsets.
SET(N, PS, PT) = subset of SET(N, PS), a valid
(*)
PT only has the change at (*), when a change happens, make the interval fixed.
PCHG(PS, PT) = count of change from BASE(PS) to PT
Example:
, changed at T1
, changed at T2
, changed at T1, T2
SUM_SUBSET(N, PS, PT) is defined in [3] = sum of all products in SET(N, PS, PT)
Now, SUM() and SUM_SUBSET() are uniformly defined as SUM(N, PS, PT), SUM(N, PS, BASE(PS)) is abbreviated as SUM(N, PS)
Only valid PT is discussed below.
[1] [2] [3] came to the following conclusion:
(1.1)
(1.2)
, PS is a Basic Shape
The following uses count of
for count of
(1.3)
,
Use the form
, Xi = Ti or Ki.
The expansion has 2M items, don’t swap the factors of
, then each
corresponds to one expression =
.
.
,
Example:
,
,
à
, K is fixed, E is variable.
, Fi = Ei or Ki
That is, a product can be broken down into 2M parts.
Define
= Sum of one part in SUM(N, PS).
PF indicates the part. Fi = Ei or Ki
Rewrite 1.3), add {braces}:
Expand SUM(N, PS, PT) by {braces}:
(1.4) SUM_K(N, PS, PT, PF) = ∑Expansion of SUM() with same
,
Example:
,
à
Expand by the {braces}:
à
In this paper, we extend the definition of Shape of Numbers and generalize the corresponding results.
2. The Extension of Shape
Redefine:
,
,
1) change factor’s domain of definition from N to Z, change K0 from 1 to Z.
2) allow
, If
, only
is allowed.
3) allow
, only
is allowed.
.
Example:
Redefine:
Basic Shape: K0 = 1 and intervals = 1 or 2
SET(N, PS) = set of items belonging to PS in [K0, N − 1], Max Factor of item ≤ N − 1
PB(PS) = Count of discontinuities in BS
PH(PS) = (Max Factor) − 1 − PB(BS)
IDX(PS) = IDX of
D1f(n): if
, then
2.1)
2.2) Specify
,
2.3)
,
, can use the form
,
,
[Proof]
Here only prove SUM(N, PS), SUM(N, PS, PT) can use the same method.
Use the similar way of [2], by definition:
(1*)
(2*)
(3*)
(4*)
(5*)
Suppose
, Max factor of PS = KM
,
1)
,
à
à Match the form
.
2)
,
à Match the form
.
3)
,
à
à Match the form
.
4)
,
By definition:
à
à Match the form
.
q.e.d.
Example:
,
à
à
;
à
à
[1, 2, 4] means
,
,
à
à
2.2. SUM_K(N, PS, PT, PF)
, K is fixed, E is variable.
,
or
That is, a product can be broken down into 2M+1 parts.
Use the same method of [2]
2.4) SUM_K(N, PS, PT, PF) is similar to (1.4), except the form =
Example:
3. Coefficient Analysis
,
Use the form
, Xi = Ti or Ki
Define
,
,
,
H(K, T, N, 1) is abbreviated as H(K, T, N)
3.1)
,
[3] has proved:
S2(M, K) is Stirling number of the second kind. à
3.2)
can use the form =
or
For arbitrary K, T:
3.3)
[Proof]
q.e.d.
this à
3.4)
can use the form:
[2] has proved:
3.5)
1.3) can derive 1.2) from this.
3.6) if
,
, then
[Proof]
Suppose
à
holds
q.e.d.
Define
, the sum traverse all combinations.
, the sum traverse all combinations.
is abbreviated as
,
is abbreviated as
;
;
;
By definition:
.
3.7) if
, then
[Proof]
Suppose H(K, T, N) holds
à
holds
q.e.d.
Example:
3.8) In
, Ki can switch the order.
3.9) if
, then
, S1 is the first kind of unsigned Stirling number.
From 3.5) and 3.7)à
à
3.10)
4.
,
, Max Factor = KM
When
à
,
or
SUM(N, PS, PT) can be broken down into 2M parts.
SUM_K(N, PS, PT, PF) can explain why SUM(N, PS, PT) has that strange form:
We can calculate every part of SUM() by some way without the form. There may be complex relationships between the parts, but their sum just match a simple form.
SUM_K(N, PS, PT, PF) use the form =
When Ti and Di all increase L times. If
, when N increase,
, match the corresponding SUM_K().
Define
SUML_K(N, PS, PF, L) = corresponding part of SUML(N, PS, L)
Above à
4.1) SUML(N, PS, L), SUML_K(N, PS, PF, L),
, can use the from
,
, 2M items in total.
,
Example:
à
à
;
4.2) P is a prime number, For arbitrary
:
If
, then
If
and
then
[Proof]
If
, then
If
and
, then
q.e.d.
5. Conclusions
The whole process of [1-3] is reviewed and this paper:
[1] tries to calculate all products of k distinct integers in [1, N − 1], introduces the concept of Shape of numbers. The idea divides all products of k distinct integers in [1, N − 1] into 2K−1 catalogs and derives the calculation formula of every catalog, that is 1.2).
[1] only introduces the basic shape. The conclusion is obtained through the derivation process.
[2] introduces the shape
, tries to calculate SUM(N, PS), the form
is guessed by observation and proved by induction.
At the same time, SUM_K() is introduced.
[3] introduces the subset, and shows the way to calculate
.
In this paper, the Shape and the form are further extended. So a lot of numbers’s series can be calculated.
Some new congruences are also obtained in [1] [2] [3] and this article.
The whole foundation is just
.