Pascal Pyramids, Pascal Hyper-Pyramids

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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
                                                                                     5
                      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