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Pascal Pyramids, Pascal Hyper-Pyramids
and a Bilateral Multinomial Theorem
Martin Erik Horn, University of Potsdam Am Neuen Palais 10, D - 14469 Potsdam, Germany E-Mail: [email protected]
Abstract
Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral Trinomial Theorem resp. a Bilateral Multinomial Theorem.
Introduction
The complete Pascal Plane with its three Pascal Triangles consists of the following numbers
(x 1 h) (y 1 h)
(x y 1 h)
lim lim
h 0 (x h)!(y h)! h 0
(x y h)!
f (x,y) + + ⋅ + +
→ →
and looks like this if the positive directions are pointed downwards:
1 1 -4 1 1 - 6 -3 1 1 -3 6 -4 3 -2 1 1 -2 3 - 1 -1 1 -1 1 1 -1 1 -1 1 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
x y
If now numbers with distance 1 are added and the definition of the bilateral hypergeometric function of [1] is used
b (b 1)
a (a 1 ) b
a 1 a 1
b 1 (a 1) (a 2)
(b 1 ) (b 2 ) 1 H 1 a;b;z ...^ z^2 z^1 z ⋅z^2 + ⋅ +
= + −^ − (2)
the following bilateral hypergeometric identity is reached:
[( y x); (y 1); 1 ] y (x y)
x x !!^1 H^1
! 2 ⋅ − + − ⋅ −
= x, y ∈ R (3)
This is a special case of the Bilateral Binomial Theorem [2, 3] with | z | = 1:
( ) [( y x);(y 1); z] (y 1) (x y 1 )
(x 1) 1 zx^ ⋅ 1 H 1 − + −
Γ Γ
Γ x, y ∈ R ; z ∈ C (4)
Part I: Pascal Pyramids and Pascal Hyper-Pyramids The Pascal Plane, which consists of binomial coefficients, can be generalized into the Pascal Space using trinomial coefficients
x!x! x!
(x x x )!
(x,x ,x )
1 2 3
1 2 3
Then the Pascal Pyramid can be constructed by adding every three appropriate neighbouring numbers and writing the result beneath them:
1
1 1 1
1 2 x 2 1 2 1
1 3 3 3 6 3 1 3 3 1 1 4 4 6 12 6 4 12 12 4 1 4 6 4 1
X 3 X 1 X 2
Remark: No, there isn’t a proud 3 sitting in the middle of the second triangle at the marked red position x. This is the place for the following humble trinomial coefficient:
3
2 3
2 3
1
3
2 3
2 3
3 2
2 3
2 3
because the construction law of trinomial coefficients reads:
(x 1 , x 2 ,x 3 ) = (x 1 −1,x 2 ,x 3 )+(x 1 ,x 2 −1,x 3 )+ (x 1 ,x 2 ,x 3 − 1) (7)
But the picture above shows only a quarter of the truth, of course, for three similar pyramids can be constructed in the negative coordinate region using these numbers
(x 1 h) (y 1 h) (z 1 h)
(x y z 1 h)
limh 0 (x h)!(y h)!(z h)! limh 0
(x y z h)!
(x,y,z)
f
→ →
as the following drawing indicates:
The next step is to increase the dimension again by considering quatronomial coefficients, which fill the four-dimensional Pascal Hyper-Space:
x! x! x !x!
(x x x x )!
(x,x ,x ,x )
1 2 3 4
1 2 3 4
(x,x ,x 1,x ) (x,x ,x ,x 1)
(x 1,x ,x ,x ) (x,x 1,x ,x )
1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4
By again using
(w 1 h) (x 1 h) (y 1 h) (z 1 h)
(w x y z 1 h)
f (w,x,y,z) limh 0 + + ⋅ + + ⋅ + + ⋅ + +
→
five Pascal Hyper-Pyramids can be found. The three-dimensional hyper-surfaces of these four-dimensional hyper-pyramids consist of the Pascal Pyramids (with some more minus- signs every now and then) sketched on the previous page.
This procedure can be continued till eternity. The multinomial coefficients
x! x !... x!
(x x ... x )!
(x,x ,...,x )
1 2 n
1 2 n
1 2 n ⋅ ⋅ ⋅
= (x 1 - 1,x 2 ,...,x 4 )+(x 1 ,x 2 - 1,...,x 4 )+...+(x 1 ,x 2 ,...,x 4 - 1) (13)
live in n-dimensional Pascal Hyper-Space, and with the help of
(x 1 h) (x 1 h)... (x 1 h)
(x x ... x 1 h) 1 2 n
1 2 n
(x,x ,...,x ) h 0
f lim
1 2 n + + ⋅ + + ⋅ ⋅ + +
→
n + 1 Pascal Hyper-Pyramids can be constructed. These n-dimensional hyper-pyramids possess (n – 1)-dimensional hyper-surfaces which look like the Pascal Pyramids of one dimension less and some more minus-signs every now and then.
Part II: Bilateral Multinomial Theorems
Formula (3) was found by adding numbers of distance 1 which lie on a straight line in the Pascal Plane. One dimension higher a similar formula should be found, if all numbers of distance 1 of the Pascal Space are added which lie in a straight plane. This then would result in powers of 3
∑ ∑^ (^ )
∞
=−∞
∞
=−∞
y x
3 n^ x;y;n x y x, y ∈ R (15)
if the series converged. But this double bilateral summation isn’t supposed to converge for it is a special case ( | z 1 | = | z 2 | = 1) of the Bilateral Trinomial Theorem
∑ ∑^ (^ )
∞
=−∞
∞
=−∞
x 2 x 1
( 1 z 1 z 2 )n^ x 1 ;x 2 ;n x 1 x 2 z 1 x^1 z 2 x^2 (16)
The Bilateral Trinomial Theorem can be reformulated as
∑ ∑
∞
=−∞
∞
=−∞ ⋅ ⋅ + −−
A A A
A
k k k
k
(n 1)
1 x y !!
x y ( )n^ (17)
where (a)k denotes the Pochhammer Symbol
x 1 , x 2 ∈ R z 1 , z 2 ∈ C
(a)
(a k) (a) (^) k Γ
Γ + = resp. (a k)
(a) (a k)k
Γ
Γ − (18)
Using the results of [2, 3] the double summation can be evaluated easily, giving a proof of formula (17) for special values of x and y.
∑ ∑ ∑ ∑
∞
=−∞
∞
=−∞
∞
=−∞
∞
=−∞ (^) − −
A (^) A A
A
A A
k A^ k A k
k
k
k
n n k
x y
x y ! !(n 1)
Of course the binomial coefficients of (19) are generalized here as
( ) ( 1) lim lim ( ) ( ) ( 1) ( 1)
! h (^)!! h
n (^) n h n h →∞ (^) h n h →∞ h n h
=^ =
+^ ⋅^ −^ +^ +^ −^ ⋅^ −^ +^ −
Γ A A A Γ A Γ A
With | x | = 1 this gives
∑ ∑ ∑
∞
=−∞
∞
=−∞
∞
=−∞ (^) − −
A (^) A A
A A
A
A A
y
x y
( 1 x)
! !(n 1)
n
k (^) n
k k^ k
∑
∞
=−∞
A
A
A (^1) x
y
( 1 x)
n n (22)
And with | y | = | 1 + x | the expected result emerges:
n n
k
k k^ k
∑ ∑ + +
∞
=−∞
∞
=−∞ − −^1 x
y
( 1 x) 1
! !(n 1)
x y A A A
A (23)
= ( 1 +x+y)^ n (24)
The same strategy leads to a Bilateral Quatronomial Theorem:
∑ ∑ ∑
∞
=−∞
∞
=−∞
∞
= −∞ ⋅ ⋅ ⋅ + −−−
m k k m k m
k m
A A A
A
(n 1)
1 x y z !!!
x y z ( )n^ (25)
with | x | = 1 , | y | = | 1 + x | and | z | = | 1 + x + y |.
And this again can be extended till eternity giving the Bilateral Multinomial Theorem:
! ) i i x ( 1 ( )
k k
1 2
k
i 1
1 xi (^) (n 1) n A A A A A (^) A
⋅∏ ∏ ∑
∞
=−∞
∞
=−∞
∞
=−∞ (^) −
i k i
with | x (^) i | = | 1 + (^) ∑
−
=
i 1
j 1
x (^) j| and.
Epilogue
To increase the aesthetical value of the indicated results a more symmetric formulation of the Bilateral Multinomial Theorem (26) can be given:
k i=
i=1 i=
Ai ∈ R x (^) i ∈ C
