Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem

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1 Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem Marti Eri Hor, Uiversit o Potsdam Am Neue Palais, D - 69 Potsdam, Germa marhor@rz.ui-potsdam.de Abstract Part I: The two-dimesioal Pascal Triagle will be geeralized ito a three-dimesioal Pascal Pramid ad our-, ive- or whatsoever-dimesioal hper-pramids. Part II: The Bilateral Biomial Theorem will be geeralised ito a Bilateral Triomial Theorem resp. a Bilateral Multiomial Theorem. Itroductio The complete Pascal Plae with its three Pascal Triagles cosists o the ollowig umbers (, ) ( + + lim lim h ( + ( + h ( ( + + ( + + ad loos lie this i the positive directios are poited dowwards: () I ow umbers with distace are added ad the deiitio o the bilateral hpergeometric uctio o [] is used (b ) (b ) b a a (a ) H + ( a; b; z)... + z + z + + z + z... (a ) (a ) a b b (b + ) + () the ollowig bilateral hpergeometric idetit is reached: [( ); ( ); ] ( ) H +, R () This is a special case o the Bilateral Biomial Theorem [, ] with z : ( + ) ( + ) ( + ) ( + z) [( ); ( + ); z] H, R ; z C () M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

2 Part I: Pascal Pramids ad Pascal Hper-Pramids The Pascal Plae, which cosists o biomial coeiciets, ca be geeralized ito the Pascal Space usig triomial coeiciets (,, ( + + ) ) (5) The the Pascal Pramid ca be costructed b addig ever three appropriate eighbourig umbers ad writig the result beeath them: X X X Remar: No, there is t a proud sittig i the middle o the secod triagle at the mared red positio. This is the place or the ollowig humble triomial coeiciet: (,, ) ( -,, ) (6) because the costructio law o triomial coeiciets reads: (,, ) (,, ) + (,, ) + (,, ) (7) But the picture above shows ol a quarter o the truth, o course, or three similar pramids ca be costructed i the egative coordiate regio usig these umbers (,, z) ( + + z + lim lim h ( + ( + (z + h as the ollowig drawig idicates: ( + + z + + ( + + ( + + (z + + (8) M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

3 z 6 6 Ad slight rotatios o the aes produce a more smmetric desig with tetrahedral order as the picture o the right is supposed to show. M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

4 The et step is to icrease the dimesio agai b cosiderig quatroomial coeiciets, which ill the our-dimesioal Pascal Hper-Space: (, B agai usig,, ( ) ) (9) (w,,, z) lim h (,,, ) + (, + (,,,,, ) ) + (, (w z + + (w + + ( + + ( + + (z + +,, ive Pascal Hper-Pramids ca be oud. The three-dimesioal hper-suraces o these our-dimesioal hper-pramids cosist o the Pascal Pramids (with some more miussigs ever ow ad the) setched o the previous page. This procedure ca be cotiued till eterit. The multiomial coeiciets (,,..., ) () () ( ) ) ()... ( -,,..., ) + (, -,..., ) (,,..., -) () live i -dimesioal Pascal Hper-Space, ad with the help o (,,..., ) lim h ( ( ( ( Pascal Hper-Pramids ca be costructed. These -dimesioal hper-pramids possess ( )-dimesioal hper-suraces which loo lie the Pascal Pramids o oe dimesio less ad some more mius-sigs ever ow ad the. Part II: Bilateral Multiomial Theorems Formula () was oud b addig umbers o distace which lie o a straight lie i the Pascal Plae. Oe dimesio higher a similar ormula should be oud, i all umbers o distace o the Pascal Space are added which lie i a straight plae. This the would result i powers o () ( ; ; ), R (5) i the series coverged. But this double bilateral summatio is t supposed to coverge or it is a special case ( z z ) o the Bilateral Triomial Theorem ( + z + z ) z z ( ; ; ) The Bilateral Triomial Theorem ca be reormulated as, R z, z C (6) ( + + ) ( + ) where (a) deotes the Pochhammer Smbol (7) M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

5 (a + ) (a) resp. (a) 5 (a) (a + ) (8) (a + ) Usig the results o [, ] the double summatio ca be evaluated easil, givig a proo o ormula (7) or special values o ad. ( + ) O course the biomial coeiciets o (9) are geeralized here as ( + ( + h ) lim lim h ( ( h + + ( + h ) ( + h ) With this gives (9) () ( + ) ( + ) ( + ) + () () Ad with + the epected result emerges: ( + ) ( + ) ( + + ) + + The same strateg leads to a Bilateral Quatroomial Theorem: () () ( z) m ( + ) m m with, + ad z + +. z m (5) Ad this agai ca be eteded till eterit givig the Bilateral Multiomial Theorem: ( + i) i ( ) + i i i i i i ( ) i (6) i with i + j ad i R. j i C Epilogue To icrease the aesthetical value o the idicated results a more smmetric ormulatio o the Bilateral Multiomial Theorem (6) ca be give: M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

6 ( ) ( + )... (7) But this o course does t chage the act that covergece is possible ol with a usmmetrical hadlig o the variables: i (8) i Literature [] William N. Baile: Series o Hpergeometric Tpe which are Iiite i Both Directios, The Quarterl Joural o Mathematics, Oord Series, Vol. 7 (96), p [] Marti E. Hor: Latacala, upublished. [] Marti E. Hor: A Bilateral Biomial Theorem, SIAM-Problem published olie at: M. Hor: Pascal Pramids, Pascal Hper-Pramids ad a Bilateral Multiomial Theorem

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie