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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

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posted: | 5/1/2010 |

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