EVOLUTIONARY MATHEMATICS AND SCIENCE FOR
GENERAL BINOMIAL COEFFICIENTS AND FIBONACCI NUMBERS
Authored by:
Hung-ping Tsao, Ph.D. 曹恆平
Tsao, Hung-ping (2021). Evolutionary mathematics and science for General Binomial Coefficients and Fibonacci Numbers. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 3, Number 4, April 2021; 34 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL3-NUM4-APR2021; ISBN 978- 0-9890870-3-2. US Department of Commerce, National Technical Information Service, 5301 Shawnee Road, Alexandria, VA 22312, USA. Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202- 4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE NEW RESEARCH REPORT 3. DATES COVERED (From - To) 4-30-2021 MAR 2021-APR 2021 4. TITLE AND SUBTITLE EVOLUTIONARY MATHEMATICS AND SCIENCE FOR 5a. CONTRACT NUMBER N/A
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9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) Wang, Lawrence K. and Tsao, Hung-ping (editors). LENOX "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Volume 3, Number 4, April 2021; Lenox Institute 11. SPONSOR/MONITOR’S REPORT NUMBER(S) STEAM-VOL3-NUM4-APR2021 Press, PO Box 405, Newtonville, NY, 12128-0405, USA
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TSAO14. ABSTRACT RESIDENCE, We first 1151generalize Highland binomial Drive, Novato, coefficients CA 94949, from USA the natural sequence based to arithmetically progressive sequences based and then display the key roles they play in the derivation of polynomial expressions of powered sums. We further come up with the new Fibonacci numbers based on arithmetically progressive sequences. As a matter of fact, the usual Fibonacci numbers are the upward diagonal sums of the Pascal triangular array, whereas these newly defined Fibonacci numbers are the upward diagonal sums of any triangular array.
15. SUBJECT TERMS (Keywords) Binomial coefficient, Pascal triangle, Fibonacci number, Natural sequence, Arithmetically progressive sequence, Stirling number, Eulerian number, Recursive formula, Upward diagonal, Fibonacci value.
16. SECURITY CLASSIFICATION OF: UNCLASSIFIED UNLIMITED 17. LIMITATION 18. NUMBER OF 19a NAME OF RESPONSIBLE (UU) OF ABSTRACT UU PAGES 34 PERSON Wang, Lawrence K. a. REPORT UU b. ABSTRACT c. THIS PAGE 19a. TELEPHONE NUMBER (include area UU UU code) (518) 250-0012
Standard Form 298 (Rev. 8-98) Prescribed by ANSI St
EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL BINOMAL COEFFICIENTS AND FIBONACCI NUMBERS
Authored by:
Hung-ping Tsao, Ph.D. 曹 恆平
3 TABLE OF CONTENTS
TABLE OF CONTENTS
ABSTRACT
KEYWORDS
NOMENCLATURE
1. INTRODUCTION
2. BINOMIAL COEFFICIENTS
3. FIBONACCI NUMBERS
GLOSSARY
REFERENCES
LIST OF TABLES
EDITORS PAGE
E-BOOK SERIES AND CHAPTER INTRODUCTON
4 EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL BINOMAL COEFFICIENTS AND FIBONACCI NUMBERS
Hung-ping Tsao 曹恆平
ABSTRACT
We first generalize binomial coefficients from the natural sequence based to arithmetically progressive sequences based and then display the key roles they play in the derivation of polynomial expressions of powered sums. We further come up with the new Fibonacci numbers based on arithmetically progressive sequences. As a matter of fact, the usual Fibonacci numbers are the upward diagonal sums of the Pascal triangular array, whereas these newly defined Fibonacci numbers are the upward diagonal sums of any triangular array.
Keywords: Binomial coefficient, Pascal triangle, Fibonacci number, Natural sequence, Arithmetically progressive sequence, Stirling number, Eulerian number, Recursive formula, Upward diagonal, Fibonacci value. 5 NOMENCLATURE
푛 (푘) binomial coefficient
푛 ( ) binomial coefficient for (a + (i −1)d)1 푘 푎;푑
Σ sum
(i)1 the natural sequence n Stirling number of the first kind k n Stirling number of the second kind k
arithmetically progressive sequence
n Stirling triangle of the first kind for k a;d
n Stirling triangle of the second kind for k a;d
n first-order Eulerian number k
n second-order Eulerian number k
T(n, k | A(i) | u;v) triangular array with infinite sequence base A(i):
∏ product
퐹푛 Fibonacci number
퐹푛(a,d) Fibonacci number for
퐹푛(푇) Fibonacci number for T
6 1. INTRODUCTION
We first generalize binomial coefficients from the natural sequence based to arithmetically progressive sequences based and then come up with new Fibonacci numbers based on arithmetically progressive sequences. As a result, significant researches and developments could be expected in both areas. We cite from (4) the following two types of arrays with any infinite sequence base A(i): T(n, k | A(i) | u;v), with T(0,0)=1, and u, v each indicating which weight to be used among W(1)=1, W(2)=A(n-1), W(3)=A(k), W(4)=A(n+k-1) +2A(1)-A(2),
W(5)=A(n-k+1), W(6)=A(k+2)-2A(1) and W(7)=A(2n-k) in the recursive formula
T(n, k)=W(u)T(n-1,k-1)+W(v)T(n-1,k).
1. Stirling: u=1, 0 ≤ 푘 ≤ 푛 and the initial values T(n, 0)=0 for n > 0. 푛 푛 푛 [ ] =T(n, k | A(i)=a+(i-1)d | 1;2), [ ] = [ ], possessing diagonal (Stirling) polynomials; 푘 푎;푑 푘 1;1 푘 푛 푛 푛 { } =T(n, k | A(i)=a+(i-1)d | 1;3), { } = { }, possessing diagonal polynomials; 푘 푎;푑 푘 1;1 푘 Lad(n, k)=T(n, k | A(i)=a+(i-1)d | 1;4), L11(n, k)=L(n, k), possessing diagonal polynomials.
2. Eulerian: u>1, −1 ≤ 푘 ≤ 푛 − 1 and T(n, 0), n > 0, varies for each array.
푛 〈 〉 =T(n, k+1 | A(i)=a+(i-1)d | 5;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0; 푘 푎;푑
푛 〈〈 〉〉 =T(n, k+1 | A(i)=a+(i-1)d | 7;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0. 푘 푎;푑
By adding W(8)=A(2)-A(1), we can obtain
푛 푛 푛 ( ) =T(n, k | A(i)=a+(i-1)d | 1;8), ( ) = ( ), possessing diagonal polynomials. 푘 푎;푑 푘 1;1 푘 7 Then the following formula from (5)
푛 푗 푛 + 1 ( ) = ∑푛 (−1)푗−푘 [ ] { } Eq. 1 푘 푗=푘 푘 푗 + 1 can further be generalized to
푛 푛 푗−푘 푗 푛 + 1 ( ) = ∑푗=푘(−1) [ ] { } . Eq. 2 푘 푎;푑 푘 푎;푑 푗 + 1 푎;푑
Thanks to the suggestion of my dear friend UNLV Professor Peter Shiue, I subsequently found
푛−1 푛−1 the following formula in (1), with ⌊ ⌋ denoting the largest integer no greater than , 2 2
푛−1 ⌊ ⌋ 푛 − 푘 − 1 퐹 = ∑ 2 ( ). Eq. 3 푛 푘=0 푘
Accordingly, I can generalize Fibonacci numbers (the natural sequence based) to the arithmetically progressive sequence based as
푛−1 ⌊ ⌋ 2 푛 − 푘 − 1 퐹푛(푎, 푑) = ∑푘=0 ( ) . Eq. 4 푘 푎;푑
Taking a=2 and d=3 for example, we can first use the recursive formula
푛 푛 − 1 푛 − 1 ( ) = ( ) +3( ) Eq. 5 푘 2;3 푘 − 1 2;3 푘 2;3
푛 and the initial values ( ) = 2푛 to first calculate the pertinent binomial coefficients and 0 2;3 then the first six Fibonacci numbers as
퐹1(2,3) = 1, 퐹2(2,3) = 2, 퐹3(2,3) = 5, 퐹4(2,3) = 13, 퐹5(2,3) = 36 and 퐹1(2,3) = 105. 8 2. BINOMIAL COEFFICIENTS
푛 Binomial coefficients (푘) can be displayed as Pascal triangle (see Table 1), which was discovered about one thousand years ago by Al-Karaji. In fact, it could trace back to the second century B.C. by Pingala and for the subsequent thousand years there had been documentary evidences that Pascal triangle had been mentioned independently in India,
Greece, China and Persia.
n\k 0 1 2 3 4 5 6 7 8 9 10
0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1
Table 1. Pascal Triangle
Nothing is more impressive than the Pascal triangle 巴斯卡三角形鮮
It displays those numbers ever so natural and simple 展示數字多自然
I have long dreamed of writing a prospective article 渴望夢想成篇寫
To show the inner beauty of numbers from my angle 數之內斂我見焉
9 First of all, let us verify Eq. 1 with Tables 1-3. n \ k 1 2 3 4 5 6 7 8 9 10 1 1 2 1 1 3 2 3 1 4 6 11 6 1 5 24 50 35 10 1 6 120 274 225 85 15 1 7 720 1764 1624 735 175 21 1 8 5040 13068 13132 6769 1960 322 28 1 9 40320 109584 118124 67284 22449 4536 546 36 1 10 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 Table 2: Table for Stirling numbers of the first kind
1 2 3 4 5 6 7 8 9 10 1 1 2 1 1 3 1 3 1 4 1 7 6 1 5 1 15 25 10 1 6 1 31 90 65 15 1 7 1 63 301 350 140 21 1 8 1 127 966 1701 1050 266 28 1 9 1 255 3025 7770 6951 2646 462 36 1 10 1 511 9330 34105 42525 22827 5880 750 45 1 Table 3. Stirling triangle of the second kind
For example, 5 2 6 3 6 4 6 5 6 ( ) = [ ] { } − [ ] { } + [ ] { } − [ ] { } = (1)(90)-(3)(65)+(11)(15)-(50)(1) =10, 2 2 3 2 4 2 5 2 6 5 3 6 4 6 5 6 ( ) = [ ] { } − [ ] { } + [ ] { } = (1)(65)-(6)(15)+(35)(1) = 10, 3 3 4 3 5 3 6 6 3 7 4 7 5 7 6 7 ( ) = [ ] { } − [ ] { } + [ ] { } − [ ] { } = (1)(350)-(6)(140)+(35)(21)-(225)(1) =20, 3 3 4 3 5 3 6 3 7 which also verified the recursive formula for binomial coefficients
10 푛 푛 − 1 푛 − 1 ( ) = ( ) + ( ). Eq. 6 푘 푘 − 1 푘
We also recall here the recursive formulas for Stirling numbers of both kinds
푛 푛 − 1 푛 − 1 [ ] = [ ] + (푛 − 1) [ ] Eq. 7 푘 푘 − 1 푘 and 푛 푛 − 1 푛 − 1 { } = { } + 푘 { }. Eq. 8 푘 푘 − 1 푘
Now let us prove Eq. 1 by mathematical induction. We shall only look at the case for n = 5 and k = 3 , since the general case is similar. We can use Eqs. 7 and 8 to show the inductive step:
6 3 6 4 6 5 6 3 6 4 6 5 6 6 6 = − + + − + − 3 3 4 3 5 3 6 3 3 3 4 3 5 3 6
2 6 4 3 6 5 4 6 6 5 6 = − − 4 + − 5 − − 6 2 3 3 3 4 3 3 5 3 3 6
6 6 4 6 6 5 6 6 6 6 = 4 + − 5 + + 6 + − 4 3 3 5 4 3 6 5 3 6
3 7 4 7 5 7 6 7 = − + − . 3 4 3 5 3 6 3 7
We then derive the following identity
k+1 n k+1 k (1+ n) = , Eq. 9 j=1 j−1 j
n where = P(n, k). k
11 n Note that the Stirling numbers of the second kind can be constructed based on = 1, 1
n n n n = = and = 1 via the inversion formula n−1 n−1 2 n
n n−k k+ j n j−1 = (−1) Eq. 10 k j=1 k k+ j as follows.
4 3 4 4 4 = − = 7 , 2 2 3 2 4
5 4 5 5 5 = − = 25 , 3 3 4 3 5
5 3 5 4 5 5 5 = − + = 15, … 2 2 3 2 4 2 5
Next, we shall use the binomial expansion
푘 (1 + 푛)푘 = ∑푘 ( ) 푛푗 Eq. 11 푗=0 푗 to prove
k+1 n k+1 k (1+ n) = . Eq. 12 j=1 j−1 j
We only look at the case where k = 4 .
From Eq. 7, we can use Eqs. 1 and 10 to write
12 4 4 4 4 4 4 2 3 4 (1+ n) = + n + n + n + n 0 1 2 3 4
1 5 2 5 3 5 4 5 = 1+ − + − n 1 2 1 3 1 4 1 5
2 5 3 5 4 5 3 5 4 5 2 3 4 + − + n + − n + n 2 3 2 4 2 5 3 4 3 5
1 5 2 2 5 3 3 3 5 2 3 2 = 1+ n + n − n + n − n + n 1 2 2 1 3 3 2 1 4
4 4 4 4 5 4 3 2 + n − n + n − n 4 3 2 1 5
n 5 n 5 n 5 n 5 n 5 = + + + + . 0 1 1 2 2 3 3 4 4 5
Next, we use the mathematical induction to prove
n k+1 n k+1 k 1 i = , Eq. 13 i=1 j=1 j j j with Eq. 12 being used in the inductive step:
n+1 k+1 n k+1 k+1 n k+1 k+1 n k+1 k 1 j 1 i = + = 1+ j j j j−1 j n − j +1 j j j i=1 j=1 j=1 j=1
k+1 n +1 n 1 k+1 k+1 1 n+1k+1 = = . Eq. 14 j=1 n − j +1 j j j j=1 j j j
Finally, we can obtain
n k k+1 j k+1 k j−k−1+r 1 k+1−r i = (−1) n Eq. 15 i=1 r=0 j=k+1−r j k+1−r j
13 by regrouping the following display of Eq. 14:
1 k+1 k+1 2 k+1 k+1 3 k+1 3 k+1 1 2 1 1 3 1 2 1 n + n − n + n − n + n + ,,, 1 1 2 2 2 1 2 3 3 3 2 3 3 1 3
k+1 k+1 k+1 k+1 k+1 k+1 1 k+1 1 k k 1 + n − n + ...+ (−1) n . k +1 k+1k+1 k +1 k k+1 k +1 1 k+1
n (a + (i −1)d) For 1 , the Stirling triangle of the first kind can be constructed via k a;d
n n−1 n−1 = +[a + (n − 2)d]) Eq.16 k a;d k−1 a;d k a;d
n n with = 1 and the Stirling triangle of the second kind can be constructed via n a;d k a;d
n n−1 n−1 = +[a + (k −1)d] Eq.17 k a;d k−1a;d k a;d
n with = 1. On the other hand, Eq.10 can be generalized in (6) to n a;d
n n−k k+ j n j−1 = (−1) . Eq. 18 k a;d j=1 k a;d k+ ja;d
n a = 2 d = 3 To elaborate, we take and . Using Eq.16, we can tabulate in Table 4. k 2;3 n \ k 1 2 3 4 5 6 7 1 1 2 2 1 3 10 7 1 4 80 66 15 1 5 880 806 231 26 1 6 12320 12164 4040 595 40 1 7 209440 219108 80844 14155 1275 57 1
Table 4. Table for general Stirling numbers of the first kind
14 n Likewise, we can use Eq. 17 to tabulate in Table 5. k 2;3
n \ k 1 2 3 4 5 6 7 1 1 2 2 1 3 4 7 1 4 8 39 15 1 5 16 203 159 26 1 6 32 1031 1475 445 40 1 7 64 5187 12831 6370 1005 57 1
Table 5. Table for general Stirling numbers of the second kind
Now, let us verify Eq. 18 for n = 5 and k = 2 :
5 3 6 4 6 5 6 = 203 = 7 159 − 66 26 + 8061 = − + . 2 2;3 2 2;3 3 2;3 2 2;3 5 2;3 2 2;3 6 2;3
Next, we shall prove
n k k+1 j−1 j k+1 k j−k−1+r d k+1−r [a + (i −1)d] = (−1) n , Eq. 19 i=1 r=0 j=k+1−r j k+1−r j a;d
n n which is the generalization of Eq. 15. By virtue of (n − j +1) = and Eq. 17, we j−1 j
first use mathematical induction to prove
k+1 k+1 n j−1 k d = (a + nd) Eq. 20 j=1 j a;d j−1
as follows. Since the inductive basis is trivially true, we only show the inductive step.
15 k+2 k+2 n j−1 d j=1 j a;d j−1
k+1 k+1 k+1 n n j−1 k+1 = d ([a + ( j −1)d] + ) + d j=1 j a;d j−1a;d j−1 k+1
k+1 k+1 n k+1 k+1 n k+1 k+1 n n j−1 j j−1 k+1 = ad + d ( j −1) + d + d j j−1 j j−1 j−1 j−1 k+1 j=1 a;d j=1 a;d j=1 a;d
k+1 k+1 n k+1 k+1 n k+1 k+1 n n j−1 j j−1 k+1 = (a + nd)d − d + d + d j=1 j a;d j−1 j=1 j a;d j j=1 j−1a;d j−1 k+1
k+1 k+1 n k+1 k+1 n k k+1 n n j−1 j j k+1 = (a + nd)d − d + d + d j=1 j a;d j−1 j=1 j a;d j j=1 j a;d j k+1
= (a + nd) k +1 .
We can now use mathematical induction to prove Eqs. 19 via 20:
k+1 d j−1 k+1 n+1 k+1 d j−1 k+1 n n = + j j=1 j j a;d j j=1 j j a;d j j−1
k+1 j−1 k+1 n k+1 k+1 n d j−1 = +d j j j j j−1 j=1 a;d j=1 a;d
n n+1 = [a + (i −1)d]k + (a + nd)k = [a + (i −1)d]k . i=1 i=1 Finally, we can obtain Eq. 19 by regrouping the following display of Eq. 20.
1 k+1 2 k+1 2 k+1 d 2 d n + n − n 1 1 2 2 2 2 1 2 a;d a;d a;d
2 3 k+1 2 3 k+1 2 3 k+1 d 3 d 2 d + n − n + n + … 3 3 3 a;d 3 2 3 a;d 3 1 3 a;d
k k+1 k+1 k k+1 k+1 k k+1 k+1 d k+1 d k k d + n − n + ...+ (−1) n . k +1 k+1k+1a;d k +1 k k+1a;d k +1 1 k+1a;d
16 Interesting readers should consult (5) for more detail. Before giving the following example,
푛 푛 푛 we would like to point out that ( ) is the generalization of ( ) , which is ( ) , 푘 푎;푑 푘 푘 1;1 from (1, 1) to (a, d), rather than from the natural sequence based to arithmetically progressive sequence based. For instance, the following recursive formula is good for any sequence 2,
2+3, x, y, z, … (Could this feature be useful in Cryptography?)
푛 푛 − 1 푛 − 1 ( ) = ( ) +3( ) 푘 2;3 푘 − 1 2;3 푘 2;3
푛 and the initial values ( ) = 2푛 , from which we can calculate Table 6. 0 2;3
n \ k 0 1 2 3 4 5 6 0 1 1 2 1 2 4 5 1 3 8 19 8 1 4 16 65 43 11 1 5 32 211 194 76 14 1 6 64 655 793 422 118 17 1 푛 Table 6. Table for ( ) 푘 2;3
We can use the difference equation method in (1) to come up with
푛 9푛2 − 15푛 + 2 ( ) = 푛 − 2 2;3 3 and 푛 4푛3 − 18푛2 + 20푛 − 3 ( ) = . 푛 − 3 2;3 3
17 3. FIBONACCI NUMBERS
We shall generalize Fibonacci numbers 퐹1 = 1, 퐹2 = 1 퐹3 = 2, 퐹4 = 3, 퐹5 = 5, 퐹6 = 8,
퐹7 = 13, 퐹8 = 21, … to be 푛−1 ⌊ ⌋ 2 푛 − 푘 − 1 퐹푛(푎, 푑) = ∑푘=0 ( ) . Eq. 21 푘 푎;푑
Taking a=2 and d=3 for example, we can use Table 6 and Eq. 22
푛−1 ⌊ ⌋ 2 푛 − 푘 − 1 퐹푛(2; 3) = ∑푘=0 ( ) . Eq. 22 푘 2;3 to calculate the first nine Fibonacci numbers as
0 1 퐹1(2,3) = ( ) = 1, 퐹2(2,3) = ( ) = 2, 0 2.3 0 2.3
2 1 3 2 퐹3(2,3) = ( ) + ( ) = 4, 퐹4(2,3) = ( ) + ( ) = 13, 0 2.3 1 2.3 0 2.3 1 2.3
4 3 2 퐹5(2,3) = ( ) + ( ) + ( ) = 36, 0 2.3 1 2.3 2 2.3
5 4 3 퐹6(2,3) = ( ) + ( ) + ( ) = 105, 0 2.3 1 2.3 2 2.3
6 5 4 3 퐹7(2,3) = ( ) + ( ) + ( ) + ( ) = 319, 0 2.3 1 2.3 2 2.3 3 2.3
7 6 5 4 퐹8(2,3) = ( ) + ( ) + ( ) + ( ) = 998, 0 2.3 1 2.3 2 2.3 3 2.3
8 7 6 5 4 퐹9(2,3) = ( ) + ( ) + ( ) + ( ) + ( ) = 3185. 0 2.3 1 2.3 2 2.3 3 2.3 4 2.3
18 As a matter of fact, the above numbers can be obtained by summing the upward diagonals of
Table 6. Likewise, the usual Fibonacci numbers can be obtained the same way from Table 1.
Therefore, the Fibonacci numbers related to any triangular array T(n, k) can be defined as
푛−1 ⌊ ⌋ 2 퐹푛(푇) = ∑푘=0 푇(푛 − 푘 − 1, 푘). Eq. 23
We can further use the generalized Fibonacci numbers to gauge the tendency of divergence of a triangular array 푇, especially when the downward diagonal polynomials of which do not
exist. For our purpose, we shall call 퐹푛(푇) of Eq. 23 the Fibonacci value of the 푛th upward diagonal of 푇.
Let us look at the triangular array t(n, k) introduced in (5) via Eq. 24
푛 + 푘 − 푗 ∑푛 푖푘 = ∑푘 푡(푛, 푗) ( ), Eq. 24 푖=1 푗=1 푘 + 1 − 푗
푛 where t(n, j) = -(n-k+1)t(n-1, j-1) + (n-k+1)t(n-1, j) and ∑푗=1 푡(푛, 푗) = 1 as follows. n\k 1 2 3 4 5 6 7 8 9 10 1 1 2 2 -1 3 6 -6 1 4 24 -36 14 -1 5 120 -240 150 -30 1 6 720 -1800 1560 -540 62 -1 7 5040 -15120 16800 -8400 1004 -126 1 8 40320 -141120 191520 -126000 37616 -3390 254 -1 9 362880 -1451520 1418480 -1905120 818080 -28624 10932 -510 1 10 362880 -45722880 22960000 -23265200 16339200 -4233520 118668 -34326 1022 -1 Table 7 Table for t(n, k)
19 Now from Table 7, we can use Eq. 23 to find the Fibonacci values 퐹푛(푡) as follows.
퐹1(푡) = 1, 퐹2(푡) = 2, 퐹3(푡) = 6 − 1 = 5, 퐹4(푡) = 24 − 6 = 18,
퐹5(푡) = 120 − 36 + 1 = 85, 퐹6(푡) = 720 − 240 + 14 = 494,
퐹7(푡) = 5040 − 1800 + 150 − 1 = 3386, 퐹8(푡) = 40320 − 15120 + 1560 − 30 = 26730,
퐹9(푡) = 362880 − 141120 + 16800 − 540 + 1 = 238021,
퐹10(푡) = 3628800 − 1451520 + 191520 − 8400 + 62 = 2360462.
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 2 5 18 85 494 3386 26730 238021 2360462
Increment Rate 2.000 2.500 3.600 4.722 5.812 6.860 7.994 8.905 9.917
Binomial coefficients have been generalized in completely different fashions such as in (2).
However, we can not use them to generalize Fibonacci numbers as we do here. In our case, we can further take d = i, where 푖 = √−1 . n \ k 0 1 2 3 4 5 6 7 8 9 10 0 1 1 2 1 2 4 2+i 1 3 8 3+2i 2+2i 1 4 16 6+3i 1+4i 2+3i 1 5 32 13+6i 2+4i -2+6i 2+4i 1 6 64 26+13i 9+8i -4+2i -6+8i 2+5i 1 7 128 51+26i 18+22i 7+4i -12-4i -11+10i 2+6i 1 8 256 102+51i 29+44i 14+29i 11-8i -22-15i -17+12i 2+7i 1 9 512 205+102i 58+80i 58i 22+40i 26-30i -34-32i -24+14i 2+8i 1 10 1024 410+205i 125+160i 80i -40+80i 52+66i 58-64i -48-56i -32+16i 2+9i 1 푛 Table 8. Table for ( ) 푘 2;푖
20 푛 We can extend the idea to the n-dimensional space, by taking into account 푖푛−1;푚 = √−푚 instead of 푖 = √−1. For example, letting 푥 = 3√−2 and 푦 = 푥2 we can look at Table 9.
n \ k 0 1 2 3 4 5 6 7 8 9 10 0 1 1 1 1 2 1 1+x 1 3 1 (1,1,1) 1+2x 1 4 1 (-1,1,1) (1,2,3) 1+3x 1 5 1 (-1,-1,1) (-7,2,3) (1,3,6) 1+4x 1 6 1 (-1,-1,-1) (-7,-8,3) (-19,3,6) (1,4,10) 1+5x 1 7 1 (3-1,-1) (-7,-8,-9) (-19,-27,6) (-39,4,10) (1,5,15) 1+6x 1 8 1 (3,3,-1) (21,-8,-9) (-19,-27,-36) (-39,-66,10) (-69,5,15) (1,6,21) 1+7x 1 9 1 (3,3,3) (21,24,-9) (93,-27,-36) (-39,-66,-102) (-69,-135,15) (-111,6,21) (1,7,28) 1+8x 1 10 1 (-5,3,3) (21,24,27) (93,117,-36) (279,-66,-102) (-69,-135,-237) (-111,-264,21) (-167,7,28) (1,8,36) 1+9x 1 Note that (-1,1,1) denote -1+1x+1y, (1,3,6) denote 1+3x+6y, … 푛 Table 9. Table for ( ) 3 푘 1; √−2
In conclusion, we are obliged to Professor H. E. Hoggatt for his formula
푛−1 ⌊ ⌋ 2 푛 − 푘 − 1 퐹푛 = ∑ ( ) 푘=0 푘 and Professor W. E. Gould for his equivalent formula
푛 ⌊ ⌋ 2 푛 − 푘 퐹푛+1 = ∑ ( ) 푘=0 푘 in (3).
We shall calculate such values of the triangular array bellow each of the following tables copied from Appendix B of (4).
21 TABLES OF VARIOUS TRIANGULAR ARRAYS
Dim n = 10, (a, d) = (1, 1), (u, v) = (1, 2), type = Stirling
n
1 1
2 1 1
3 2 3 1
4 6 11 6 1
5 24 50 35 10 1
6 120 274 225 85 15 1
7 720 1764 1624 735 175 21 1
8 5040 13068 13132 6769 1960 322 28 1
9 40320 109584 118124 67284 22449 4536 546 36 1
10 362880 1026576 1172700 723680 269325 63273 9450 870 45 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 1 3 9 36 176 1030 7039 55098 486346
Increment Rate 1.000 3.000 3.000 4.000 4.889 5.852 6.834 7.828 8.827
Dim n = 10, (a, d) = (2, 3), (u, v) = (1, 2), type = Stirling
n
1 1
2 2 1
3 10 7 1
4 80 66 15 1
5 880 806 231 26 1
6 12320 12164 4040 595 40 1
7 209440 219108 80844 14155 1275 57 1
8 4188800 4591600 1835988 363944 39655 2415 77 1
9 96342400 109795600 46819324 10206700 1276009 95200 4186 100 1
10 2504902400 2951028000 1327098024 312193524 43382934 3751209 204036 6786 126 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 2 11 87 947 13141 221836 4411974 101015440 2616548183
Increment Rate 2.000 5.500 7.909 10.885 13.896 16.881 19.888 22.896 25.903
22 Dim n = 10, (a, d) = (1, 1), (u, v) = (1, 3), type = Stirling
n
1 1
2 1 1
3 1 3 1
4 1 7 6 1
5 1 15 25 10 1
6 1 31 90 65 15 1
7 1 63 301 350 140 21 1
8 1 127 966 1701 1050 266 28 1
9 1 255 3025 7770 6951 2646 462 36 1
10 1 511 9330 34105 42525 22827 5880 750 45 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 1 2 4 9 22 58 164 495 1587
Increment Rate 1.000 2.000 2.000 2.250 2.444 2.636 2.828 3.018 3.206
Dim n = 10, (a, d) = (2, 3), (u, v) = (1, 3), type = Stirling
n
1 1
2 2 1
3 4 7 1
4 8 39 15 1
5 16 203 159 26 1
6 32 1031 1475 445 40 1
7 64 5187 12831 6370 1005 57 1
8 128 25999 107835 82901 20440 1974 77 1
9 256 130123 888679 1019746 369061 53998 3514 100 1
10 512 650871 7239555 12105885 6186600 1287027 124278 5814 126 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 2 5 15 56 250 1255 6816 39532 244880
Increment Rate 2.000 2.500 3.000 3.733 4.464 5.020 5.431 5.800 6.194
23 Dim n = 10, (a, d) = (1, 1), (u, v) = (1, 4), type = Stirling
n
1 1
2 2 1
3 6 6 1
4 24 36 12 1
5 120 240 120 20 1
6 720 1800 1200 300 30 1
7 5040 15120 12600 4200 630 42 1
8 40320 141120 141120 58800 11760 1176 56 1
9 362880 1451520 1693440 846720 211680 28224 2016 72 1
10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 2 7 30 157 972 6961 56660 516901 5225670
Increment Rate 2.000 3.500 4.286 5.233 6.191 7.162 8.140 9.123 10.110
Dim n = 10, (a, d) = (2, 3), (u, v) = (1, 4), type = Stirling
n
1 1
2 4 1
3 28 14 1
4 280 210 30 1
5 3640 3640 780 52 1
6 58240 72800 20800 2080 80 1
7 1106560 1659840 592800 79040 4560 114 1
8 24344320 42602560 18258240 3043040 234080 8778 154 1
9 608608000 1217216000 608608000 121721600 11704000 585200 15400 200 1
10 17041024000 38342304000 21909888000 5112307200 589881600 36867600 1293600 25200 252 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 4 29 294 3851 61910 1180141 26025012 651805441 18276577360
Increment Rate 4.000 7.250 10.138 13.099 16.076 19.062 22.052 25.045 28.040
24 Dim n = 10, (a, d) = (1, 1), (u, v) = (5, 6), type = Euler
n
1 1
2 1 1
3 1 4 1
4 1 11 11 1
5 1 26 66 26 1
6 1 57 302 302 57 1
7 1 120 1191 2416 1191 120 1
8 1 247 4293 15619 15619 4293 247 1
9 1 502 14608 88234 156190 88234 14608 502 1
10 1 1013 47840 455192 1310354 1310354 455192 47840 1013 1
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 1 2 5 13 38 125 449 1742 7269
Increment Rate 1.000 2.000 2.500 2.600 3.167 3.289 3.592 3.880 4.173
Dim n = 10, (a, d) = (1, 1), (u, v) = (7, 6), type = Euler
n
1 1
2 1 2
3 1 8 6
4 1 22 58 24
5 1 52 328 444 120
6 1 114 1452 4400 3708 720
7 1 240 5610 32120 58140 33984 5040
8 1 494 19950 195800 644020 785304 341136 40320
9 1 1004 67260 1062500 5765500 12440064 11026296 3733920 362880
1 0 1 2026 218848 5326160 44765000 155357384 238904904 162186912 44339040 3628800
Diagonal Number 1 2 3 4 5 6 7 8 9 10
Fibonacci Value 1 1 3 9 29 111 467 2157 10625 56783
Increment Rate 1.000 3.000 3.000 3.222 3.828 4.207 4.576 4.926 5.344
25 GLOSSARY
Polynomial: A mathematical expression such as ax3+bx2-cx, where x is a variable and a, b, c are called coefficients.
Binomial expansion: According to the binomial theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form a xb yc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b.
Combinatorics: The branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints.
Mathematical induction: To prove a statement S(n) is true for any natural number n, it suffices first to establish the inductive basis [to prove S(1) is true] and then to provide the inductive step [to prove S(m+1) is true by assuming S(m) is true].
Fibonacci numbers: They are commonly denoted as Fn, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
퐹1 = 1, 퐹2 = 1 퐹3 = 2, 퐹4 = 3, 퐹5 = 5, 퐹6 = 8, 퐹7 = 13, 퐹8 = 21, …
26 REFERENCES
1. Hoggatt Jr., V. E., and Lind D. A., “FIBONACCI AND BINOMIAL PROPERTIES OF WEIGHTED COMPOSITIONS”, Journal of Combinatorial Theory 4, 121-124 , Elsevier, 1968.
2. H. W. Gould, “THE BRACKET FUNCTION AND FONTENE-WARD GENERALIZED BINOMIAL COEFFICIENTS WITHAPPLICATION TO FIBONOMIAL COEFFICIENTS”, Fibonacci Quarterly, Vol. 7, No 1, 381-391, 1969.
3. H. W. Gould, “THE CASE OF THE STRANGE BINOMIAL IDENTITIES OF PROFESSOR MORIARTY”, Fibonacci Quarterly, Vol. 10, No 1, 23-40, 1972.
4. Tsao H., and Chang L., “EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL”, Lenox Institute Press, Newtonville, , New York, USA, 2021.
5. Tsao H., “EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL POWERED SUMS OF NUMBERS: STIRLING-EULER-LAH-BELL”, Lenox Institute Press, Newtonville, , New York, USA, 2021.
6. Tsao H., “EXPLICIT POLYNOMIAL EXPRESSIONS FOR SUMS OF POWERS OF AN ARITHMETIC PROGRESSION”, Mathematical Gazette, Cambridge, England, 2008.
27 LIST OF TABLES
Table 1. Pascal Triangle
Table 2. Table for Stirling numbers of the first kind
Table 3. Table for Stirling numbers of the second kind
n Table 4. Table for general Stirling numbers of the first kind k 2;3
n Table 5. Table for general Stirling numbers of the second kind k 2;3
푛 Table 6. Table for ( ) 푘 2;3
Table 7 Table for t(n, k)
푛 Table 8. Table for ( ) 푘 2;푖
푛 Table 9. Table for ( ) 3 푘 1; √−2
28 Editors of "EVOLUTIONARY PROGRESS IN SCIENCE, TECHNOLOGY,
ENGINEERING, ARTS AND MATHEMATICS (STEAM)"
1. Dr. Lawrence K. Wang (王 抗 曝 )
Lawrence K. Wang has over 30+ years of professional experience in facility design, environmental sustainability, natural resources, STEAM education, global pollution control, construction, plant operation, and management. He has expertise in water supply, air pollution control, solid waste disposal, drinking water treatment, waste treatment, and hazardous waste management.
He was the Director/Acting President of the Lenox Institute of Water Technology, Engineering
Director of Krofta Engineering Corporation and Zorex Corporation, and a Professor of
RPI/SIT/UIUC, in the USA.
He was also a Senior Advisor of the United Nations Industrial and Development
Organization (UNIDO) in Austria.
Dr. Wang is the author of over 700 technical papers and 45+ books, and is credited with 24 US patents and 5 foreign patents.
29 He earned his two HS diplomas from the High School of National Taiwan Normal University and the State University of New York. He also earned his BS degree from National Cheng-
Kung University, Taiwan, ROC, his two MS degrees from the University of Missouri and the
University of Rhode Island, USA, and his PhD degree from Rutgers University, USA.
Currently he is the Chief Series Editor of the Handbook of Environmental Engineering series
(Springer); Chief Series Editor of the Advances in Industrial and Hazardous Wastes Treatment series, (CRC Press, Taylor & Francis); co-author of the Water and Wastewater Engineering series (John Wiley & Sons); and Co-Series Editor of the Handbook of Environment and
Waste Management series (World Scientific). Dr. Wang is active in professional activities of
AWWA, WEF, NEWWA, NEWEA, AIChE, ACS, OCEESA, etc.
2. Dr. Hung-ping Tsao (曹恆平)
Hung-ping Tsao has been a mathematician, a university professor, and an assistant actuary, serving private firms and universities in the United States and Taiwan for 30+ years. He used to be an Associate Member of the Society of Actuaries and a Member of the American Mathematical Society. His research have been in the areas of college mathematics, actuarial mathematics, management mathematics, classic number theory and Sudoku puzzle solving.
30 In particular, bikini and open top problems are presented to share some intuitive insights and some type of optimization problems can be solved more efficiently and categorically by using the idea of the boundary being the marginal change of a well-rounded region with respect to its inradius; theory of interest, life contingency functions and pension funding are presented in more simplified and generalized fashions; the new way of the simplex method using cross- multiplication substantially simplified the process of finding the solutions of optimization problems; the generalization of triangular arrays of numbers from the natural sequence based to arithmetically progressive sequences based opens up the dimension of explorations; the introduction of step-by-step attempts to solve Sudoku puzzles makes everybody’s life so much easier and other STEAM project development.
Dr. Tsao is the author of 3 books and over 30 academic publications. Among all of the above accomplishments, he is most proud of solving manually in the total of ten hours the hardest
Sudoku posted online by Arto Inkala in early July of 2012.
He earned his high school diploma from the High School of National Taiwan Normal
University, his BS and MS degrees from National Taiwan Normal University, Taipei,
Taiwan, his second MS degree from the UWM in USA, and a PhD degree from the
University of Illinois, USA.
31
Editors of the eBOOK Series of the "EVOLUTIONARY PROGRESS IN
SCIENCE, TECHNOLOGY, ENGINEERING, ARTS AND MATHEMATICS
(STEAM)"
Dr. Lawrence K. Wang (王 抗 曝 ) - - left
Dr. Hung-ping Tsao (曹 恆 平) -- right
32 E-BOOK SERIES AND CHAPTER INTRODUCTON
Introduction to the E-BOOK Series of the "EVOLUTIONARY PROGRESS IN SCIENCE,
TECHNOLOGY, ENGINEERING, ARTS AND MATHEMATICS (STEAM)" and This
Chapter “EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL BINOMIAL COEFFICIENTS AND FIBONACCI NUMBERS”
The acronym STEM stands for “science, technology, engineering and mathematics”. In accordance with the National Science Teachers Association (NSTA), “A common definition of STEM education is an interdisciplinary approach to learning where rigorous academic concepts are coupled with real-world lessons as students apply science, technology, engineering, and mathematics in contexts that make connections between school, community, work, and the global enterprise enabling the development of STEM literacy and with it the ability to compete in the new economy”. The problem of this country has been pointed out by the US Department of Education that “All young people should be prepared to think deeply and to think well so that they have the chance to become the innovators, educators, researchers, and leaders who can solve the most pressing challenges facing our nation and our world, both today and tomorrow. But, right now, not enough of our youth have access to quality STEM learning opportunities and too few students see these disciplines as springboards for their careers.” STEM learning and applications are very popular topics at present, and STEM related careers are in great demand. According to the US Department of Education reports that the number of STEM jobs in the United States will grow by 14% from 2010 to 2020, which is much faster than the national average of 5-8 % across all job sectors. Computer programming and IT jobs top the list of the hardest to fill jobs.
33 Despite this, the most popular college majors are business, law, etc., not STEM related. For this reason, the US government has just extended a provision allowing foreign students that are earning degrees in STEM fields a seven month visa extension, now allowing them to stay for up to three years of “on the job training”. So, at present STEM is a legal term. The acronym STEAM stands for “science, technology, engineering, arts and mathematics”. As one can see, STEAM (adds “arts”) is simply a variation of STEM. The word of “arts” means application, creation, ingenuity, and integration, for enhancing STEM inside, or exploring of STEM outside. It may also mean that the word of “arts” connects all of the humanities through an idea that a person is looking for a solution to a very specific problem which comes out of the original inquiry process. STEAM is an academic term in the field of education.
The University of San Diego and Concordia University offer a college degree with a STEAM focus. Basically STEAM is a framework for teaching or R&D, which is customizable and functional, thence the “fun” in functional. As a typical example, if STEM represents a normal cell phone communication tower looking like a steel truss or concrete column, STEAM will be an artificial green tree with all devices hided, but still with all cell phone communication functions. This e-book series presents the recent evolutionary progress in STEAM with many innovative chapters contributed by academic and professional experts.
This e-book chapter, “EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL BINOMIAL COEFFICIENTS AND FIBONACCI NUMBERS” is Dr. Hung-ping Tsao’s collection of thoughts, works and articles about various ways of coming up with formulas for sums of powers throughout his retired period for seventeen years now. Three years prior to the publication of “EXPLICIT POLYNOMIAL EXPRESSIONS FOR SUMS OF POWERS OF AN ARITHMETIC PROGRESSION”, he gave a few talks among universities in Taiwan and a class of gifted students of his Alma Mater (High School of National Taiwan Normal University). He was then invited to present “General Triangular Arrays of Numbers” by “22nd Asian Technology Conference in Mathematics” (Chung Yuan Christian University, December 19, 2017). He is also grateful that Professor Ronald Graham [author of “CONCRETE MATHEMATICS”] replied promptly to my e-mails with two separate attachments of his manuscripts that he generalized most of the special functions in Chapter 6 of “CONCRETE MATHEMATICS”. He is presenting here a systemic but rather long account of his personal excursion into the realm of numbers initiated by Blaise Pascal, James Stirling, Leonhard Euler and Jacob Bernoulli, which is therefore not meant to be a categorical survey of the topic.
34