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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: marhorn@rz.uni-potsdam.de
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 + y + h)!                             Γ(x + y + 1 + h)                                    (1)
f            =   lim                          =            lim
(x, y)        h →0     (x + h)!⋅ (y + h)!              h →0   Γ(x + 1 + h) ⋅ Γ(y + 1 + h)
and looks like this if the positive directions are pointed downwards:

1                                                      1
-4          1                                         1            -4
6           -3         1                              1            -3         6
-4        3           -2         1                     1        -2           3          -4
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

1 H1    (a; b; z ) = ... + (b − 1) ⋅ (b − 2) ⋅ z −2 + b − 1 ⋅ z −1 + 1 + a ⋅ z + a ⋅ (a + 1) ⋅ z 2 + ...                   (2)
(a − 1) ⋅ (a − 2)                a −1                b        b ⋅ (b + 1)
the following bilateral hypergeometric identity is reached:
x!
2x =                    ⋅ H [( y − x ); (y + 1); − 1]                                                   x, y ∈ R          (3)
y !⋅ (x − y) ! 1 1
This is a special case of the Bilateral Binomial Theorem [2, 3] with | z | = 1:
Γ(x + 1)
(1 + z )x =                              ⋅ H [( y − x ); (y + 1); − z ]                            x, y ∈ R ; z ∈ C       (4)
Γ(y + 1) ⋅ Γ(x − y + 1) 1 1
M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem
2
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
(x1 + x 2 + x 3 )!
(x1 , x 2 , x 3 ) =                                                                                                      (5)
x1! ⋅ x 2 ! ⋅ x 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

X3
X1                                                                                                 X2

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:

( 2 , 2 , 2) =
3 3 3                2 2 2
2!
(3
= 3⋅ - 1 , 2 , 2
3 3
)                                                         (6)
!⋅ !⋅ !
3 3 3

because the construction law of trinomial coefficients reads:
(x1 , x 2 , x 3 ) = (x1 − 1, x 2 , x 3 ) + (x1 , x 2 − 1, x 3 ) + (x1 , 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 + y + z + h)!                                            Γ(x + y + z + 1 + h)                (8)
f               =   lim                                     =                 lim
(x, y, z)       h →0       (x + h)!⋅ (y + h)!⋅ (z + h)!                   h →0     Γ(x + 1 + h) ⋅ Γ(y + 1 + h) ⋅ Γ(z + 1 + h)
as the following drawing indicates:

M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem
3

y

20                    140
10                60                  210
4                20                    60
1                                                                 1                4                 10                     20

10                    60
6                 30                  90
–3                                                                   3                12                    30
3           1                                                     1                3                 6                      10

4                     20
–3                                                                     3                 12                  30
6                  2        1                                        2                6                     12
–3          –2                    –1                              1                2                 3                      4
z
–1                                                                 1                     4
3                   1     –1                                           1                 3                   6
–3                 –2       –2     –1        2                       1                2                     3
1           1                     1               1               1                1                 1                      1      x
1
1              –1
1              –2                  1
1                  –3              3                 –1
1

–1
1                –1

1
–2               2
1                –2                1

–1
3                3
–3               6                     –3
1                –3                3                      –1

And slight rotations of the axes produce a
more symmetric design with tetrahedral
order as the picture on the right is supposed
to show.

M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem
4
The next step is to increase the dimension again by considering quatronomial coefficients,
which fill the four-dimensional Pascal Hyper-Space:
(x1 + x 2 + x 3 + x 4 )!
(x1 , x 2 , x 3 , x 4 ) =                                                                                                           (9)
x1! ⋅ x 2 ! ⋅ x 3! ⋅ x 4 !
= (x1 − 1, x 2 , x 3 , x 4 ) + (x1 , x 2 − 1, x 3 , x 4 )
(10)
+ (x1 , x 2 , x 3 − 1, x 4 ) + (x1 , x 2 , x 3 , x 4 − 1)

By again using
Γ(w + x + y + z + 1 + h)                                                    (11)
f                  =       lim
(w, x, y, z)           h →0         Γ(w + 1 + h) ⋅ Γ(x + 1 + h) ⋅ Γ(y + 1 + h) ⋅ Γ(z + 1 + h)
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
(x1 + x 2 + ... + x n )!
(x1 , x 2 , ..., x n ) =                                                                                                           (12)
x1! ⋅ x 2 ! ⋅ ... ⋅ x n !
= (x1 - 1, x 2 , ..., x 4 ) + (x1 , x 2 - 1, ..., x 4 ) + ... + (x1 , x 2 , ..., x 4 - 1) (13)

live in n-dimensional Pascal Hyper-Space, and with the help of
Γ(x 1 + x 2 + ... + x n + 1 + h)                                           (14)
f                          =    lim
(x , x ,...,x )                 h →0      Γ(x 1 + 1 + h) ⋅ Γ(x 2 + 1 + h) ⋅ ... ⋅ Γ(x n + 1 + h)
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
∞        ∞
3n =        ∑ ∑ (x; y; n − x − y)                                                                           x, y ∈ R               (15)
y = −∞ x = −∞

if the series converged. But this double bilateral summation isn’t supposed to converge for it
is a special case ( | z1 | = | z2 | = 1) of the Bilateral Trinomial Theorem
∞            ∞                                                         x 1, x 2 ∈ R
(1 + z1 + z 2 ) n =         ∑ ∑ (x ; x                  1     2   ; n − x 1 − x 2 )⋅ z1
x1
⋅ z2
x2
z1, z2 ∈ C              (16)
x = −∞ x = −∞
2            1

The Bilateral Trinomial Theorem can be reformulated as
∞            ∞                   xk ⋅ y
(1 + x + y) n =            ∑ k∑ k !⋅ !⋅ (n + 1)                                                                                    (17)
=−∞ =−∞              −k −

where (a)k denotes the Pochhammer Symbol

M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem
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Γ(a + k)                                                            Γ(a)
(a) k =                            resp.              (a + k) − k =                                                          (18)
Γ(a)                                                              Γ(a + k)
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.
∞        ∞             xk ⋅ y                                   ∞ n ∞ n −                         k               
∑ k∑ k !⋅ !⋅ (n + 1)                                  =      ∑    ⋅ k∑  k
       
= −∞    =−∞ 
⋅x ⋅ y
                 
   (19)
= −∞ =−∞             −k −                                                                                              
Of course the binomial coefficients of (19) are generalized here as

n              (n + h) !                     Γ(n + h − 1)
  = lim                        = lim                                                                                       (20)
    h →∞ ( + h) ! ⋅ (n − + h)!   h →∞ Γ( + h − 1) ⋅ Γ(n − + h − 1)
With | x | = 1 this gives
∞        ∞             xk ⋅ y                               ∞            n                                
∑ k∑ k !⋅ !⋅ (n + 1)                                 =       ∑            ⋅ (1 + x ) n− ⋅ y
                                
             (21)
= −∞ =−∞             −k −                                    =−∞                                           
∞    n  y                    
= (1 + x ) n ⋅             ∑   ⋅ 1+ x 
 
=−∞        


(22)
                             
And with | y | = | 1 + x | the expected result emerges:
n
∞        ∞             xk ⋅ y                                        y 
∑ k∑ k !⋅ !⋅ (n + 1)
= −∞ =−∞
= (1 + x ) ⋅ 1 +   
 1+ x 
n
(23)
−k −
= (1 + x + y) n                                                        (24)

The same strategy leads to a Bilateral Quatronomial Theorem:
∞           ∞        ∞                x k ⋅ y ⋅ zm
(1 + x + y + z) n =               ∑ ∑ ∑                                                                                      (25)
m=−∞ =−∞ k = −∞                  k !⋅ !⋅ m ! ⋅ (n + 1) −k −               −m

with | x | = 1 , | y | = | 1 + x | and | z | = | 1 + x + y | .

And this again can be extended till eternity giving the Bilateral Multinomial Theorem:

∞            ∞                   ∞                                k                   k
k
1
(1 + ∑ x i ) n =          ∑ ∑ ••• ∑                                                            ⋅∏x         i       ∏( i!)    (26)
i =1                =−∞ =−∞ =−∞                                     (n + 1)    k         i=1     i           i=1
1            2                   k                          −∑
i=1   i

i−1
∈R
with | x i | = | 1 +    ∑1x   j   | and                    i
.
j=                             xi ∈ C

Epilogue
To increase the aesthetical value of the indicated results a more symmetric formulation of the
Bilateral Multinomial Theorem (26) can be given:

M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem
6
(x 0 + x1 + x 2 + ... + x k ) n
        1  0           1  1                1                  k
∞       ∞      ∞    x0 −        ⋅  x1 −      ⋅ ... ⋅  x k −                               (27)
      k +1           k +1                k +1
=    ∑ ∑ •••
= −∞ =−∞
∑
=−∞           ! ⋅ !⋅... ⋅ ! ⋅ (n + 1) − − −...−
0       1          k                0       1          k               0   1     k

But this of course doesn’t change the fact that convergence is possible only with an
unsymmetrical handling of the variables:

1        1                 1
x i = 1 + x1 −      + x 2 − + ... + x i −1 −                                                                  (28)
k        k                 k

Literature
[1] William N. Bailey: Series of Hypergeometric Type which are Infinite in Both Directions,
The Quarterly Journal of Mathematics, Oxford Series, Vol. 7 (1936), p. 105 – 115.
[2] Martin E. Horn: Lantacalan, unpublished.
[3] Martin E. Horn: A Bilateral Binomial Theorem,
SIAM-Problem published online at: http://www.siam.org/journals/problems/03-001.htm

M. Horn: Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem

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