JURNAL FOURIER | Oktober 2024.
Vol.
No.
2, 111-117
DOI: 10.
14421/fourier.
ISSN 2252-763X
E-ISSN 2541-5239
Generalized Gaussian Fibonacci Numbers and its Determinantal Identities Yashwant Kumar Panwar1.
Vinod Kumar Gupta2.
Minal Gwala3 Department of Mathematics.
PMCoE.
Rajiv Gandhi.
Government P.
College.
Mandsaur.
INDIA Department of Mathematics.
Government Madhav Science P.
College.
Ujjain.
INDIA Corresponding Author: Yashwant Kumar Panwar.
Email: yashwantpanwar@gmail.
Abstract In this paper, we present the determinantal identities of generalized Gaussian Fibonacci numbers.
The generalized Gaussian Fibonacci sequence ycyeNyea ya .
ec, yee.
yeC, yeE) is defined by the recurrence relation ycyeNyea ya = yecycyeNyea ya yeeycyeNyea , with ycyeNya = yeC and ycyeNya = yeE.
This was introduced by S.
Pethe and A.
Horadam.
Also, we present its determinantal identities with classical numbers like gaussian Fibonacci.
Lucas.
Pell.
Pell-Lucas.
Jacobsthal, jacobsthal-Lucas.
Bronze.
Nickel and Mersenne numbers.
Keywords: Gaussian Fibonacci number.
Gaussian Lucas number.
Gaussian Pell number.
Gaussian Pell-Lucas number.
Gaussian Jacobsthal number.
Gaussian jacobsthal-Lucas number and determinant.
Introduction Determinants have played a significant part in various areas in mathematics.
For instance, they are quite useful in the analysis and solution of system of linear equations.
There are different perspectives on the study of determinants.
Complex Fibonacci numbers .
, often known as Gaussian Fibonacci numbers, were first presented by Horadam .
In 1965.
Jordan .
studied two sequences of complex numbers and came up with certain traits of regular Fibonacci sequences.
Gaussian Fibonacci numbers .
, are defined by for ycu Ou 2, yayaycu = yayaycuOe1 yayaycuOe2 With yaya0 = ycn, yaya1 = 1.
One can see that yayaycu = yaycu ycnyaycuOe1 Where yaycu is the nth Fibonacci number.
Gaussian Lucas numbers .
, are defined by for ycu Ou 2, yayaycu = yayaycuOe1 yayaycuOe2 With yaya0 = 2 Oe ycn, yaya1 = 1 2ycn.
One can see that yayaycu = yaycu ycnyaycuOe1
A 2024 JURNAL FOURIER
Versi online via w.
Yashwant K Panwar1.
Gupta2.
Minal Gwala3 Where yaycu is the nth Lucas number.
The Gaussian Fibonacci.
Gaussian Lucas.
Gaussian Pell.
Gaussian PellAeLucas.
Gaussian Jacobsthal.
Gaussian JacobsthalAeLucas.
Gaussian Bronze.
Gaussian Nickel, and Gaussian Mersenne numbers are examples of new complex number sequences that are created by combining recursively defined numerical sequences with Gaussian type integers.
In this paper, we present generalized Gaussian Fibonacci numbers and its determinantal identities.
We also establish results in terms of gaussian Fibonacci numbers, gaussian Lucas numbers, gaussian Pell numbers, gaussian Pell-Lucas numbers, gaussian Jacobsthal numbers, gaussian jacobsthal-Lucas numbers.
Generalized Gaussian Fibonacci Sequence Definition 1.
The generalized Gaussian Fibonacci sequence .
, 5, .
, is defined by the recurrence yayceycu 2 = ycyyayceycu 1 ycyayceycu with yayce0 = yca and yayce1 = yca, where yca ycaycuycc yca are initial values.
First few generalized Gaussian Fibonacci numbers are yayce2 = ycyyca ycyca yayce3 = ycy2 yca ycyycyca ycyca yayce4 = ycy3 yca ycy2 ycyca 2ycyycyca yc 2 yca yayce5 = ycy4 yca ycy3 ycyca 3ycy2 ycayc 2ycy2 ycyca ycyycyca If yca = ycn, yca = 1, ycy = 1, yc = 1, then Gaussian Fibonacci numbers yayaycu = yayaycu yayaycuOe1 If yca = ycn, yca = 1, ycy = 2, yc = 1, then Gaussian Pell numbers .
yaycEycu = 2yaycEycu yaycEycuOe1 If yca = ycn/2, yca = 1, ycy = 1, yc = 2, then Gaussian Jacobsthal numbers .
yayaycu = yayaycu 2yayaycuOe1 If yca = ycn, yca = 1, ycy = 3, yc = 1, then Gaussian Bronze numbers .
, .
yayaAycu = 3yayaAycu yayaAycuOe1 If yca = ycn/3, yca = 1, ycy = 1, yc = 3, then Gaussian Nickel numbers .
yaycAycu = yaycAycuOe1 3yaycAycuOe2 ycn If yca = Oe 2 , yca = 1, ycy = 3, yc = Oe2, then Gaussian Mersenne numbers .
JURNAL FOURIER .
13 111-117 Generalized Gaussian Fibonacci numbers and its .
yaycAycu = 3yaycAycu Oe 2yaycAycuOe1 If yca = 2 Oe ycn, yca = 1 2ycn, ycy = 1, yc = 1, then Gaussian Lucas numbers .
yayaycu = yayaycu yayaycuOe1 If yca = 2 Oe 2ycn, yca = 2 2ycn, ycy = 2, yc = 1, then Gaussian Pell-Lucas numbers .
yaycEycu = 2yaycEycu yaycEycuOe1 If yca = 2 Oe ycn/2, yca = 1 2ycn, ycy = 1, yc = 2, then Gaussian Jacobsthal-Lucas numbers .
yaycycu = yaycycu 2yaycycuOe1 Determinantal Identities Determinants of matrices with generalized fibonaci entries are present on research .
In this discussion, some results are given about the determinant of matrices related to the generalized Gaussian Fibonacci number that has been defined previously.
Theorem 2.
If yayceycu is the generalized Gaussian Fibonacci numbers, then .
c yayceycu 1 ycyyayceycu 2 yc 2 yayceycu 1 yayceycu 3 ycyayceycu 1 ycy2 yayceycu 2 ycy yayceycu 2 yayceycu 3 Proof.
Let OI= .
c yayceycu 1 ycyyayceycu 2 yc 2 yayceycu 1 yayceycu 3 ycyayceycu 1 yayceycu 3 | = 2yc 3 yayceycu 1 ycy3 yayceycu 2 yayceycu 3 ycyyayceycu 2 yayceycu 3 ycyayceycu 1 ycy2 yayceycu 2 ycy yayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyyayceycu 2 yayceycu 3 Assume ycyayceycu 1 = yeC and ycyyayceycu 2 = yeE, then by the definition of generalized Gaussian Fibonacci sequence yayceycu 3 = yeC yeE, now .
OI= | yeC2 yeE yeC2 .
eC yeE) yeCyeE2 yeE .
eC yeE) yeC.
eC yeE)2 yeE.
eC yeE)2 | Taking yeC, yeE and .
eC yeE) ycayc ycaycuycoycoycuycu yceycycuyco ycI1 , ycI2 , ycI3 ycyceycycyyceycaycycnycyceycoyc OI= yeCyeE.
eC yeE) .
eC2 yeC2 yeE2 yeE2 .
eC yeE)2 .
eC yeE)2 | OI= yeCyeE.
eC yeE) | 0 yeC2 yeE2 OeyeE2 yeE2 .
eC yeE)2 .
eC yeE)2 | Applying ycI2 Ie ycI2 Oe ycI3 .
JURNAL FOURIER .
13 111-117 Yashwant K Panwar1.
Gupta2.
Minal Gwala3 Expanding by ycI3 , yeE2 OI= yeC yeE.
eC yeE) | 2 OeyeE .
eC yeE)2 .
eC yeE)2 OI= yeC3 yeE.
eC yeE).
eE2 .
eC yeE)2 yeE2 .
eC yeE)2 ] OI= yeC3 yeE3 .
eC yeE)3 Put ycyayceycu 1 = yeC, ycyyayceycu 2 = yeE and yayceycu 3 = yeC yeE, we get OI= 2yc 3 yayceycu 1 ycy3 yayceycu 2 yayceycu 3 This completes the Proof.
Corollary 3.
If yca = ycn, yca = 1, ycy = 1, yc = 1, then Gaussian Fibonacci numbers .
ayaycu 1 yayaycu 2 yayaycu 1 yayaycu 3 yayaycu 1 yayaycu 2 yayaycu 2 yayaycu 3 yayaycu 1 yayaycu 3 | = 2yayaycu 1 yayaycu 2 yayaycu 3 yayaycu 2 yayaycu 3 Corollary 4.
If yca = ycn, yca = 1, ycy = 2, yc = 1, then Gaussian Pell numbers .
aycEycu 1 2yaycEycu 2 yaycEycu 1 yaycEycu 3 yaycEycu 1 4yaycEycu 2 4yaycEycu 2 yaycEycu 3 yaycEycu 1 yaycEycu 3 | = 16yaycEycu 1 yaycEycu 2 yaycEycu 3 2yaycEycu 2 yaycEycu 3 Corollary 5.
If yca = ycn/2, yca = 1, ycy = 1, yc = 2, then Gaussian Jacobsthal numbers .
yayaycu 1 yayaycu 2 4yayaycu 1 yayaycu 3 2yayaycu 1 yayaycu 2 yayaycu 2 yayaycu 3 2yayaycu 1 yayaycu 3 | = 16yayaycu 1 yayaycu 2 yayaycu 3 yayaycu 2 yayaycu 3 Corollary 6.
If yca = ycn, yca = 1, ycy = 3, yc = 1, then Gaussian Bronze numbers .
ayaAycu 1 3yayaAycu 2 yayaAycu 1 yayaAycu 3 yayaAycu 1 9yayaAycu 2 9yayaAycu 2 yayaAycu 3 yayaAycu 1 yayaAycu 3 | = 54yayaAycu 1 yayaAycu 2 yayaAycu 3 yayaAycu 2 yayaAycu 3 Corollary 7.
If yca = ycn/3, yca = 1, ycy = 1, yc = 3, then Gaussian Nickel numbers .
yaycAycu 1 yaycAycu 2 9yaycAycu 1 yaycAycu 3 3yaycAycu 1 yaycAycu 2 yayceycu 2 yayceycu 3 3yaycAycu 1 yaycAycu 3 | = 54yaycAycu 1 yaycAycu 2 yaycAycu 3 yaycAycu 2 yaycAycu 3 ycn Corollary 8.
If yca = Oe 2 , yca = 1, ycy = 3, yc = Oe2, then Gaussian Mersenne numbers JURNAL FOURIER .
13 111-117 Generalized Gaussian Fibonacci numbers and its .
yaycAycu 1 3yaycAycu 2 4yaycAycu 1 yaycAycu 3 Oe2yaycAycu 1 9yaycAycu 2 9yaycAycu 2 yaycAycu 3 Oe2yaycAycu 1 yaycAycu 3 | = Oe432yaycAycu 1 yaycAycu 2 yaycAycu 3 3yaycAycu 2 yaycAycu 3 Corollary 9.
If yca = 2 Oe ycn, yca = 1 2ycn, ycy = 1, yc = 1, then Gaussian Lucas numbers .
ayaycu 1 yayaycu 2 yaya2ycu 1 yayaycu 3 yayaycu 1 yaya2ycu 2 yayaycu 2 yayaycu 3 yayaycu 1 yaya2ycu 3 yayaycu 2 yaya2ycu 3 | = 2yaya3ycu 1 yaya3ycu 2 yaya3ycu 3 Corollary 10.
If yca = 2 Oe 2ycn, yca = 2 2ycn, ycy = 2, yc = 1, then Gaussian Pell-Lucas numbers .
c yayceycu 1 ycyyayceycu 2 yc 2 yayceycu 1 yayceycu 3 ycyayceycu 1 ycy2 yayceycu 2 ycy yayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 | = 2yc 3 yayceycu 1 ycy3 yayceycu 2 yayceycu 3 ycyyayceycu 2 yayceycu 3 Corollary 11.
If yca = 2 Oe ycn/2, yca = 1 2ycn, ycy = 1, yc = 2, then Gaussian Jacobsthal-Lucas numbers .
yaycycu 1 yaycycu 2 4yaycycu 1 yaycycu 3 2yaycycu 1 yaycycu 2 yaycycu 2 yaycycu 3 2yaycycu 1 yaycycu 3 | = 16yaycycu 1 yaycycu 2 yaycycu 3 yaycycu 2 yaycycu 3 The proof of the theorems 12 to 18, are in line with the proof of Theorem 2.
Theorems 2, 12, 13, 14, and 15 use determinant-based identities to uncover profound and diverse structural characteristics of extended Gaussian Fibonacci numbers.
They give a robust foundation for generalizing such results to other number sequences and algebraic systems.
Theorem 12.
If yayceycu is the generalized Gaussian Fibonacci numbers, then .
cyyayceycu 2 yayceycu 3 }2 yc 2 yayceycu 1 yc 2 yayceycu 1 .
cyayceycu 1 yayceycu 3 }2 ycy2 yayceycu 2 ycy2 yayceycu 2 .
cyayceycu 1 ycyyayceycu 2 } yayceycu 3 yayceycu 3 = 3ycyycyayceycu 1 yayceycu 2 yayceycu 3 .
cyayceycu 1 ycyyayceycu 2 yayceycu 3 }3 Theorem 13.
If yayceycu is the generalized Gaussian Fibonacci numbers, then ycyayceycu 1 ycyyayceycu 2 ycyyayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyayceycu 1 ycyyayceycu 2 | | ycyyayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyayceycu 1 ycyyayceycu 2 ycyyayceycu 2 yayceycu 3 = 6ycyycyayceycu 1 yayceycu 2 yayceycu 3 Oe 2[.
cyayceycu 1 }3 .
cyyayceycu 2 }3 .
ayceycu 3 }3 ] Theorem 14.
If yayceycu is the generalized Gaussian Fibonacci numbers, then JURNAL FOURIER .
13 111-117 Yashwant K Panwar1.
Gupta2.
Minal Gwala3 2yayceycu 3 ycyayceycu 1 ycyyayceycu 2 ycyayceycu 1 yayceycu 3 2ycyayceycu 1 ycyyayceycu 2 yayceycu 3 yayceycu 3 ycyayceycu 1 = 2.
cyayceycu 1 ycyyayceycu 2 yayceycu 3 }3 ycyyayceycu 2 ycyyayceycu 2 2ycyyayceycu 2 ycyayceycu 1 yayceycu 3 Theorem 15.
If yayceycu is the generalized Gaussian Fibonacci numbers, then ycyyayceycu 2 yayceycu 3 ycyayceycu 1 ycyyayceycu 2 ycyayceycu 1 ycyyayceycu 2 yayceycu 3 ycyyayceycu 2 | | yayceycu 3 ycyayceycu 1 ycyayceycu 1 ycyyayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 = yc yayceycu 1 ycy yayceycu 2 yayceycu 3 Oe 3ycyycyayceycu 1 yayceycu 2 yayceycu 3 Theorems 16, 17 and 18 arrive at the same elegant result, highlighting a deep consistency in the behavior of determinants constructed from these sequences.
This reinforces the strong algebraic structure and recurrence symmetry inherent in generalized Gaussian Fibonacci numbers.
Theorem 16.
If yayceycu is the generalized Gaussian Fibonacci numbers, then ycy2 yayceycu 2 yayceycu 3 | ycyycyayceycu 1 yayceycu 2 ycyayceycu 1 ycyycyayceycu 1 yayceycu 2 yc 2 yayceycu 1 yayceycu 3 ycyyayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyyayceycu 2 yayceycu 3 | = .
ycyycyayceycu 1 yayceycu 2 yayceycu 3 }2 yc 2 yayceycu 1 ycy2 yayceycu 2 Theorem 17.
If yayceycu is the generalized Gaussian Fibonacci numbers, then Oeyc 2 yayceycu 1 .
cyycyayceycu 1 yayceycu 2 ycyayceycu 1 yayceycu 3 ycyycyayceycu 1 yayceycu 2 Oeycy2 yayceycu 2 ycyyayceycu 2 yayceycu 3 ycyayceycu 1 yayceycu 3 ycyyayceycu 2 yayceycu 3 | = .
ycyycyayceycu 1 yayceycu 2 yayceycu 3 }2 Oeyayceycu 3 Theorem 18.
If yayceycu is the generalized Gaussian Fibonacci numbers, then yc 2 yayceycu 1 ycyyayceycu 2 yayceycu 3 .
c yayceycu 1 ycyycyayceycu 1 yayceycu 2 ycy2 yayceycu 2 ycyycyayceycu 1 yayceycu 2 ycy2 yayceycu 2 ycyyayceycu 2 yayceycu 3 = .
ycyycyayceycu 1 yayceycu 2 yayceycu 3 } yayceycu 3 ycyayceycu 1 yayceycu 3 ycyayceycu 1 yayceycu 3 yayceycu 3 Acknowledgments The author thanks the editor and anonymous referees for helpful suggestions and comments.
Conclusion This paper developed determinant identities of generalized Gaussian Fibonacci numbers in generalized Also results derived in terms of classical Gaussian numbers like Fibonacci.
Lucas.
Pell.
Pell-Lucas.
Bronze.
Nickel, and Mersenne numbers.
We have obtained recursive results in all determinantal These identities can be used to developed new identities for classical polynomials.
These JURNAL FOURIER .
13 111-117 Generalized Gaussian Fibonacci numbers and its .
findings are also in line with previous studies on determinants of matrices using general Fibonacci-type References