JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2016.41020JAMP-63190ArticlesPhysics&Mathematics <i>N</i>-Summet-<i>k</i> and Its Application in the Construction of Pascal Triangle and Pascal Matrix eelamJeevan Kumar1*Department of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University Hyderabad College of Engineering, Jagtial, India* E-mail:neelamjeevankumar@engineer.com11012016040116917717 December 2015accepted 26 January 29 January 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet- k. The result of variation between N and k is shown in tabulation.

Summetor <i>N</i>-Summet-<i>k</i> Binomial Coefficients Pascal’s Triangle Pascal’s Matrix
1. Introduction

Summetor was firstly introduced in research article, “Jeevan-Kushalaiah Method to Find the Coefficients of Characteristic Equation of a Matrix and Introduction of Summetor” by the authors Neelam Jeevan Kumar and Neelam Kushalaiah  . The Summetor operator name is taken from “sum” operator. Summetor operation is “Sum of all positive integers form one to n”. N-summet-k: Sum of all positive integers summeted k-times progressively.

N-summet-k is (Figure 1).

Representation of N-summet-k. Symbol and explanation: N is the real number or complex number. It must symbolize in Uppercase english letter only. k is the real number. It must symbolize in lowercase english letter only. <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-1720466x8.png" xlink:type="simple"/></inline-formula> is Summetor operator symbol. Tale end “+” represents “summation”

Example:

Properties:

2. Tabulation and Graph2.1. Tabulation

In the given Table 1, N is taken on vertically and k is taken on horizontally. The result of N-summet k is given with variable N from −9 to 15 and variable k from −3 to 10. N is positive value. The tabulation is the heart of N- summet-k.

In Table 2, elbow arrow between 6.4 and 10 proves

N-summet-k values, where N = −9 to 0 and k = −3 to 9
−3−2−1N↓/k↔+1+2+3+4+5+6+7+8+9
001−936−84126−12684−369−10
001−828−5670−5628−8100
001−721−3535−217−1000
001−615−2015−610000
001−510−105−100000
001−46−41000000
001−33−10000000
001−2100000000
001−1000000000
0010000000000

N (vertical bold numbers) and k (horizontal bold numbers).

N-summet-k values, where N = 1 to 15 and k = −3 to 9
0011111111111
001261015212836455566
00141020355684120165220286
00151535701262103304957151001
00162156126252462792128720023003
001728842104629241716300350058008
00183612033079217163432643511,44019,448
00194516549512873003643512,87024,31043,758
00110552207152002500511,44024,31048,62092,378
001116628610013003800819,44843,75892,378184,756
00112783641365436812,37631,82475,582167,960352,716
00113914551820618818,56450,388125,970293,930646,646
001141055602380856827,13277,520203,490497,4201,144,066
00115120680306011,62838,760116,280319,770817,1901,961,256

But most commonly used formula to calculate N-summet-k is

and

The dotted inclined lines shows Pascal triangle

2.2. Graph

Note: Taking = 0, for -1 > k,

where R > 0 and x is any integer or number or equation (Figures 2-4).

Proof-1: N < 0, N is any non fractional real number. Assume N = −1.From Equation (i.c) we get

Observe the tabulation; the N-summet-k value is 0.

 Proof-2: k < -1, let k = −2.

From Equations (i.a) and (i.c) gives

Observe the tabulation, the N-summet-k value is 0.

3. Applications3.1. Pascal’s Triangle (<xref ref-type="fig" rid="fig5">Figure 5</xref>)

Pascal’s triangle  is a triangular array of the binomial coefficients   . Binomial coefficients are indexed

by two non negative integers n and k written as. It is the Coefficient of term in polynomial expression of. Where n rises from 0 to n.

The coefficients are given by the expression. k varies from 0 to n.

By using Summetor or N-summet-k can be written as

where k varies from 0 to n in R.H.S and r varies from −1 to n−1 in L.H.S

A set, S has n-elements. The number of k combinations can be calculated by using Equation (ii)

Equation (ii) also gives combinations i.e., nCk, k varies from 0 to n formulated with N-summet-k.

The computational time taken to calculate when, k > 0 is much higher than that of (n − k)k

The computation taken to calculate is t1, 1 < k < n and

The computation taken to calculate

3.2. Pascal’s Matrix

The Pascal matrix  -  is an n × n dimension infinite matrix containing the binomial coefficients as its elements. The Pascal matrix generation is the matrix exponential of a special subdiagonal or superdiagonal matrix. The Three Pascal Matrices are Upper Triangular Matrix (Un), Lower Triangular Matrix (Ln) and Symmetric Matrix (Sn). Symmetric Matrix is product of Lower Triangular Matrix and Upper Triangular Matrix. The m is the values of Subdiagonal or superdiagonal elements and lies between −∞ and +∞.

9 × 9 Pascal Matrices (Un, Ln and Sn) represented rows as i = n = 9 and column as j = n = 9.

For positive values of diagonal elements, the Pascal matrices are

3.2.1. Upper Triangular Matrix, (U<sub>n</sub>)

Upper triangular matrix is formatted with exponential matrix containing superdiagonal elements.

Elements of upper triangular matrix are

3.2.2. Lower Triangular Matrix, (L<sub>n</sub>)

Lower triangular matrix is formatted with exponential matrix containing subdiagonal elements.

Elements of lower triangular matrix are

Lower triangular matrix is transpose of upper triangular matrix vice versa.

3.2.3. Symmetric Matrix, (S<sub>n</sub>)

Elements of Sn for positive values of subdiagonal or superdiagonal elements.

For positive values subdiagonal or superdiagonal elements

The (Square box) in tabulation shows 12 × 12 Pascal Symmetric Matrix for positive values and;

The in tabulation shows mirror image elements positions of 5 × 5 Pascal upper triangular matrix.

Acknowledgements

N-summet-k has numerous applications in mathematics and physics like simplification of laguerre polynomials   applied in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one- electron atom and Calculation of electrical voltage distribution across high voltage suspension type string insulator  and grading of string insulators  to improve string efficiency of high voltage overhead transmission line and so on. Author found the application of N-summet-k in above three applications but the subject regarding these applications not published yet. In future, the above three applications will be published by using N-summet-k.

Cite this paper

Neelam JeevanKumar, (2016) N-Summet-k and Its Application in the Construction of Pascal Triangle and Pascal Matrix. Journal of Applied Mathematics and Physics,04,169-177. doi: 10.4236/jamp.2016.41020

ReferencesKumar, N.J. and Kushalaiah, N. (2013) Jeevan-Kushalaiah Method to Find the Coefficients of Characteristic Equation of a Matrix and Introduction of Summetor. International Journal of Scientific & Engineering Research, 4, 1553-1562.http://dx.doi.org/10.14299/ijser.2013.08.002Conway, J.H. and Guy, R.K. (1996) Pascal’s Triangle. In: Conway, J.H. and Guy, R.K., Eds., The Book of Numbers, Springer-Verlag, New York, 68-70. http://dx.doi.org/10.1007/978-1-4612-4072-3Coolidge, J.L. (1949) The Story of the Binomial Theorem. The American Mathematical Monthly, 56, 147-157.http://dx.doi.org/10.2307/2305028Flower, D. (1996) The Binomial Coefficient Function. The American Mathematical Monthly, 103, 1-17.http://dx.doi.org/10.2307/2975209Edwards, A.W.F. (2013) The Arithmetical Triangle. In: Wilson, R. and Watkins, J.J., Eds., Combinatorics: Ancient and Modern, Oxford University Press, Oxford, 166-180. http://dx.doi.org/10.1093/acprof:oso/9780199656592.003.0008Helms, G. (2006-2008) Pascal Matrix in a Project of Compilation of Facts about Binomial Related Matrices. http://go.helms-net.de/math/binomial/index.htmEdelman, A. and Strang, G. (2004) Pascal Matrices. American Mathematical Monthly, 111, 361-385. http://dx.doi.org/10.2307/4145127Call, G.S. and Velleman, D.J. (1993) Pascal’s Matrices. American Mathematical Monthly, 100, 372-376.http://dx.doi.org/10.2307/2324960Sequence of Pascal Traces in OEIS, A006134. https://oeis.org/A006134/listSpain, B. and Smith, M.G. (1970) Functions of Mathematical Physics. Van Nostrand Reinhold Company, London, Chapter 10 Deals with Laguerre Polynomials.Hansen, P., Massey, W. and Chavez, J. (2006) Numerical Calculation of Voltage Distribution in an Insulator String Comparison with Measurements. IEEE Antennas and Propagation Society International Symposium, Albuquerque, 9-14 July 2006, 2929-2932. http://dx.doi.org/10.1109/APS.2006.1711220Denholm, A.S. (1960) Electric Stress Grading of Insulator Strings. Electrical Engineering, 79, 647. http://dx.doi.org/10.1109/EE.1960.6432764