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A Novel Approach of Determining Stella Octangula Number And International Journal of Statistics and Applied Mathematics 2017; 2(6): 261-263 ISSN: 2456-1452 Maths 2017; 2(6): 261-263 © 2017 Stats & Maths A novel approach of determining Stella Octangula www.mathsjournal.com Received: 03-09-2017 number and Pronic number using initial value theorem Accepted: 04-10-2017 in Z - Transform G Janaki Department of Mathematics, Cauvery College for Women, G Janaki and S Vidhya Trichy-18, Tamil Nadu, India S Vidhya Abstract Department of Mathematics, In this communication, we evaluate Stella Octangula number and Pronic number by applying the initial Cauvery College for Women, value theorem in Z – Transform. Trichy-18, Tamil Nadu, India Keywords: Inverse Z – Transform, Pronic number, Stella Octangula number Notations 2 SO n 2 n 1 n = Stella Octangula number of rank n . Pro n n 1 n = Pronic number of rank n . Introduction Life is full of patterns; especially Mathematics is also full of patterns. The main aim of Number theory is to discover interesting and unexpected relationships. In [1-7], theory of numbers was discussed. The Z – Transform is useful for the manipulation of discrete date sequences and has acquired a new significance in the formulation and analysis of discrete-time systems. It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, and economics. These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations. In [8-10], Z-Transform methods were analysed. Recently in [11], the sequence of m-gonal numbers and octahedral numbers was developed. In this communication, we evaluate Stella Octangula number and Pronic number by using initial value theorem in Z-Transform. Definition u n 0,1, 2, u 0 If the function n is defined for discrete values and n for n 0 , then n Z u U ( z ) u z its Z – Transform is defined to be n n . The inverse Z – Transform 1 Z U ( z ) u n . is written as Initial Value Theorem u Lt U ( z ) Z (u ) U ( Z ), 0 n z If then . Method of Analysis The process of finding the Stella Octangula number and Pronic number by using initial value Correspondence theorem in Z – Transform is given in the following theorems. S. Vidhya Department of Mathematics, Cauvery College for Women, Trichy-18, Tamil Nadu, India ~261~ International Journal of Statistics and Applied Mathematics Theorem 1: z 3 10 z 2 z Z 1 SO 4 n z 1 Proof Assume that z 3 10 z 2 z U ( z ) . 4 z 1 By initial value theorem, we have u 0 Lt U ( z ) 0 z z 3 10 z 2 z u Lt z U ( z ) u Lt z 0 1 1 0 4 z z ( z 1) z 3 10 z 2 z 1 u Lt z 2 U ( z ) u u z 1 Lt z 2 14 2 2 2 2 1 2 0 1 4 z z ( z 1) z z 3 10 z 2 z 1 14 u Lt z 3 U ( z ) u u z 1 u z 2 Lt z 3 51 3 2 3 2 1 3 0 1 2 4 2 z z ( z 1) z z z 3 10 z 2 z 1 14 51 u Lt z 4 U ( z ) u u z 1 u z 2 u z 3 Lt z 4 124 4 2 4 2 1 4 0 1 2 3 4 2 3 z z ( z 1) z z z 5 1 2 3 4 u 5 Lt z U ( z ) u 0 u 1 z u 2 z u 3 z u 4 z z z 3 10 z 2 z 1 14 51 124 Lt z 5 245 5 2 5 2 1 4 2 3 4 z z 1 z z z z Continuing the above process, we get n 1 2 3 ( n 1) u n Lt z U ( z ) u 0 u 1 z u 2 z u 3 z u n 1 z z z 3 10 z 2 z 1 14 51 Lt z n 4 2 3 z z 1 z z z u n 2 n 2 1 SO n n Thus, Theorem 2: 2 z 2 Z 1 Pro 3 n z 1 Proof Assume that 2 z 2 U ( z ) . 3 z 1 By initial value theorem, we have 2 z 2 u Lt z U ( z ) u Lt z 0 2 11 1 1 0 3 z z ( z 1) ~262~ International Journal of Statistics and Applied Mathematics 2 z 2 2 u Lt z 2 U ( z ) u u z 1 Lt z 2 6 2 2 1 2 0 1 3 z z ( z 1) z 2 z 2 2 6 u Lt z 3 U ( z ) u u z 1 u z 2 Lt z 3 12 33 1 3 0 1 2 3 2 z z ( z 1) z z 2 z 2 2 6 12 u Lt z 4 U ( z ) u u z 1 u z 2 u z 3 Lt z 4 20 4 4 1 4 0 1 2 3 3 2 3 z z ( z 1) z z z 5 1 2 3 4 u 5 Lt z U ( z ) u 0 u 1 z u 2 z u 3 z u 4 z z 2 z 2 2 6 12 20 Lt z 5 30 5 5 1 3 2 3 4 z z 1 z z z z Continuing the above process, we get n 1 2 3 ( n 1) u n Lt z U ( z ) u 0 u 1 z u 2 z u 3 z u n 1 z z 2 z 2 2 6 12 Lt z n 3 2 3 z z 1 z z z u n n 1 Pro n n Thus, Conclusion In this communication, we find Stella Octangula number and Pronic number by applying the initial value theorem in Z – Transform. To conclude that, one can find the other special numbers by using various properties of Z – Transform. References 1. Carmichael RD. The Theory of numbers and Diophantine Analysis, Dover Publications, New York, 1959. 2. Mordell LJ. Diophantine equations, Academic Press, London, 1969. 3. Nagell T. Introduction to Number Theory, Chelsea Publishing Company, New York, 1981. 4. Hua LK. Introduction to Theory of Numbers, Springer-Verlag, Berlin-New York, 1982. 5. Oistein Ore. Number Theory and its History, Dover, New York, 1988. 6. David Wells. The Penguin Dictionary of curious and interesting numbers, Penguin Book, 1997. 7. Dickson LE. History of Theory of Numbers, 2, Diophantine Analysis, New York, Dover, 2005. 8. Eliahu Ibrahim Jury. Theory and Application of the Z-Transform method, John Wiley and Sons. P.L, 1964. 9. Eliahu Ibrahim Jury. Theory and Application of the Z-Transform method, Krieger Pub Co. ISBN 0-88275-122-0, 1973. 10. Kanasewich ER. Time sequence analysis in geophysics (3rd ed.), University of Alberta, 185-186, 1981. 11. Pandichelvi V, Sivakamasundari P. An innovative approach of evaluating Polygonal numbers and Octahedral Number, International Journal of Development Research, 2017; 7(5):12940-12943. 2 z 2 Z 1 Pro 3 n z 1 ~263~ .
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