CC BY
7Probl. Anal. Issues Anal. Vol. 13 (31), No3, 2024, pp. 43-55
DOI: 10.15393/j3.art.2024.16070
43
UDC 517.521
J. HOSSAIN, S. DEBNATH
ON X^-CONVERGENCE OF SEQUENCES OF BI-COMPLEX NUMBERS
Abstract. We propose the concept of X^-convergence of sequences of bi-complex numbers. We explore the fundamental properties of this newly introduced notion and its relationships with other convergence methods.
Key words: bi-complex number, ideal, filter, X-convergence, I* -convergence, X^-convergence.
2020 Mathematical Subject Classification: 40A35, 40G15.
1. Introduction. Ideal convergence, originally proposed by Kostyrko et al. [7] in 2001 as an extension of statistical convergence, has garnered significant attention from researchers in subsequent years. Starting from this foundational work, researchers (e.g.) Debnath and Rakshit [6], Choudhury and Debnath [3], Savas and Das [13], among others, have conducted extensive research in this area, exploring its applications and properties. Through their investigations, they have highlighted ideal convergence as a generalized form encompassing various established convergence concepts, contributing to the advancement of mathematical analysis and its applications. For an extensive study on Ideal convergence, one may refer to [9], [10], [13].
In 2011, Macaj and Sleziak [8] proposed the concept of XK-convergence, which extends the idea of X*-convergence by incorporating two ideals, X and K. Unlike the traditional convergence, where convergence is assessed along a single set, X^-convergence considers convergence along a large set with respect to another ideal, K. This extension presents an intriguing analogy and offers avenues for further exploration. For an extensive study on I^-convergence, one may refer to [4].
The notion of XK-convergence being a generalization of I*-convergence suggests potential for deeper investigation and application. Recent advancements in this direction, particularly from a topological perspective,
© Petrozavodsk State University, 2024
have been made by Debnath et al. [5] and other researchers. Their works shed light on the topological aspects of X^-convergence, contributing to a broader understanding of this generalized convergence concept and its implications in various mathematical contexts.
The exploration of convergence is a fundamental aspect of analysis, playing a crucial role in various mathematical investigations. However, the study of convergence of sequences of bi-complex numbers remains relatively underdeveloped and has not yet received substantial attention. Despite its nascent stage, recent research indicates a notable analogy in the convergence behavior of sequences of bi-complex numbers.
Recently, Bera and Tripathy [1] made significant strides by introducing the concept of statistical convergence for sequences of bi-complex numbers. Their work marks a pivotal advancement in the study of convergence in this domain, as they explored the different properties from both algebraic and topological perspectives.
Given this recent progress, it is indeed a natural progression to explore the Z* -convergence of sequences of bi-complex numbers. Building upon the foundation laid by Bera and Tripathy, investigating X^-convergence offers the opportunity to deepen our understanding of the convergence behavior of bi-complex sequences in a more generalized setting. This exploration may unveil new insights into the convergence properties of bi-complex numbers, contributing to the broader landscape of mathematical analysis.
Throughout the paper, C2 represent the set of all bi-complex numbers.
2. Definitions and Preliminaries. Segre [15] defined a bi-complex number as £ = z\ + i2z2 = x\ + i\x2 + i2x3 + i\i2x4, where z\ = x\ + %\x2, z2 = x3 + i2x4 e C (set of complex numbers) and X\, X2, X3, X4 e R (set of real numbers) and the independent units i1,i2 are such that i2 = i"2 = —1 and i\i2 = i2i\. Denote the set of all bi-complex numbers by C2; it is defined as: C2 = : £ = z1 + i2z2: z1,z2 e C}.
In the realm of bi-complex numbers, a number £ = x1 + i1x2 + i2x3 + + i\i2x4 is classified as a hyperbolic number if x2 = 0 and x3 = 0. These hyperbolic numbers are collectively denoted as V., and the set comprising them is referred to as the V-plane.
Equipped with coordinate-wise addition, real scalar multiplication, and term-by-term multiplication, the set C2 becomes a commutative algebra with the identity 1 = 1 + i\ ■ 0 + i2 ■ 0 + i\i2 ■ 0.
Within C2, there exist four idempotent elements, specifically 0, 1,
1 + H%2 1 - i\i2
ei =-^-, and &2 =-^-.
It is obvious that ei + e2 = 1 and eie2 = e2ei = 0. Every bi-complex number £ = zi + i2z2 has a unique idempotent representation as £ = Tiei + T2e2, where Ti = zi — iiz2 and T2 = zi + iiz2 are called the idempotent components of £. The Euclidean norm || • || on C2 is defined as
IICHca = = Vi^ii2 + i*212 = \JlTil2 + lT22,
where £ = zi + i2z2 = xi + iix2 + i2x3 + iii2x4 = Tiei + T2e2 with this norm C2 is a Banach space. For an extensive view, [11], [14], [16] can be addressed where many more references can be found.
Definition 1. [2] In the context of bi-complex numbers, several conjugates are defined as follows:
The ii-conjugate of a bi-complex number £ = zi + i2z2 is denoted by £ * and is defined as £* = zi + i2z2 for all zi,z2 e C. Here, zi and z2 represent the complex conjugates of zi and z2, respectively, and ii = i"2 = — 1.
The i2-conjugate of a bi-complex number £ = zi + i2z2 is denoted by £ and is defined as £ = zi — i2z2 for all zi,z2 e C. Again, ii = i2 = — 1.
The i\i2-conjugate of a bi-complex number £ = z\ + i2z2 is denoted by £ and defined as £ = zi — i2z2 for all z\,z2 e C, where z\ and z2 are the complex conjugates of Z\ and z2, respectively, and i"2 = i"2 = — 1.
Properties of i\-conjugation. Some of the properties of i\-conjugation, which were obtained by Rochon and Shapiro [12] are listed as follows:
(i) (£ + v)* = £ * + v*;
(ii) «)* = at*;
(iii) (D* = 6
(iv) m* = £V;
(v) (C^* = (e*)"iif £^ exists;
(vi)
* — L
They are obtained similarly to the properties of i2 -conjugation.
Definition 2. [2] A sequence of bi-complex number ) is considered statistically convergent to £ e C2 if, for every £ > 0,
8({k e N: ||& — e|c2 ^ e})= 0.
Symbolically, we write stat-lim =
Definition 3. [7] Consider a nonempty set X. A family of subsets X c V (X) is called an ideal on X if it satisfies the following conditions:
(1) For every X1 ,X2 e X, the union X1 y X2 belongs to X.
(2) For every X1 e X and every subset X2 of X1,X2 is also in X.
Further X is said to be admissible if @x e X,{x} e X, and it is said to be nontrivial if X ^ 0 and X R X.
Example 1. Here are some standard examples of ideals:
(1) The collection of all finite subsets of N constitutes a nontrivial admissible ideal on N, denoted as Xf.
(2) The set comprising all subsets of N with natural density zero forms a nontrivial admissible ideal on N. This particular ideal is denoted as X$.
(3) Let Xc = {A c N: £ a~1 < 8} c N. Then Xc forms an ideal in N.
aeA
(4) Consider a partitioning of the natural numbers N into disjoint sets
co
D1,D2, D3,..., such that N = \J Dp and Da x Db = 0 for a ^ b. The set
p=i
X, comprising all subsets of N that have finite intersections with the sets Dp, constitutes an ideal in N.
Definition 4. [3] A family T c 2X of subsets of a nonempty set X is called a filter in X if it satisfies the following conditions:
(1) The empty set does not belong to T.
(2) For all Xi,X2 e T, the intersection Xi x X2 is also in T.
(3) For every X1 e T and every superset X2 of X1 containing X1,X2 is also in T.
Definition 5. [3] If X is a proper nontrivial ideal in Y, then T(X) = {A c Y: DB e X: A = Y — B} constitutes a filter in Y. This filter is commonly referred to as the filter associated with the ideal X.
Definition 6. [7] Let X c P(N) denote a nontrivial ideal over N. We define an X-convergence for a real-valued sequence (£n) towards I as follows: for every £ > 0, the set H(e) = {n e N : — l\ ^ e} must be an element of X. Here, I is called the X-limit of the sequence (£n) and is denoted by X — lim Cfc " l.
Definition 7. [4] Consider an admissible ideal X in N. Define X*-convergence for a real-valued sequence ) towards I as follows: there exists a
set T = {tl < t2 < ... < tk < ...} in the associated filter T(X) such that lim£k = I. Symbolically, X* — lim= ¡.
keT k
Definition 8. [8] Let X and K be two ideals in N. A real-valued sequence (£k) is said to be XK-convergent to l if there exists M e T(X), such that
the sequence (rqk) defined by rqk = i ' ' is K-convergent to ¡.
I ¡, k R M,
Definition 9. [8] Let K be an ideal on N. Then P cK Q denotes the property P\Q e K. Also, P cK Q and Q cK P together imply P Q. Thus, P „k. Q if and only if PAQ e K. A set P is said to be IC-pseudo intersection of a system {Pi: i e N} if for every i e N, P cK Pi holds.
Definition 10. [8] Let X and K be two ideals on N. Then X is said to have the additive property with respect to K (or the condition AP(X, K,) holds), if every sequence (Fn)neN of sets from T(X) has IC-pseudo intersection in T (X).
Definition 11. With respect to the Euclidean norm on C2, a sequence of bi-complex numbers (£k) is called X-convergent to t e C2 if, for each £l > 0, the set
F(£i) = {k e N : \\£k — t\\C2 > £i} e X.
X— ll'llco
Symbolically, we write, -> t.
Definition 12. Let X be an admissible ideal. A sequence ) of C2 is called X*-convergent to £ e C2 with respect to the Euclidean norm on C2 if D a set T = {tl < t2 < ... < tk < ...} in the associated filter T(X), such that the sub-sequence (£tk) converges to Symbolically, we write,
, X*—Hlc2 Kk-> K.
3. Main Results.
Definition 13. Let X and K be two admissible ideals in N and ) be a sequence of bi-complex numbers. Then (£k) is considered XK-convergent to I e C2 with respect to the Euclidean norm on C2 if there exists
M e T(X), such that the sequence (rqk) defined by rqk = i ' '
I I, k R M,
is K-convergent to I. Symbolically, we write, -¡.
8
Example 2. Let Dp = {2p~1k: k is an odd number}. Then, N = [J Dp
p=i
is a decomposition of N. Let X = Xd , those subsets of N that intersect with
finitely many D„'s. Consider the sequence ) defined by ^ = —--,
@ k e Dp's. Then the sequence is Xx-convergent to 0.
Justification. Let M = N\ D1. Then M e T(X). Now, consider the
sequence (rjk) defined by rjk = | ^, ^ ^ m
Now, the sequence (rjk) = (0, -1——2, 0, -1——2, 0,...) is X-convergent
2 3
to 0.
If e > -, then A(e) = {k e N: || rjk — 0||C2 ^ £} = 0e X, as 0 intersects none of Dp's. If e = -, then A(-) = {k e N: ||rjk — 0||c2 ^ -} = D- e X, as it intersects only D2. If £ = 1, then A( 1) = {k e N: ||rjk — 0||C2 ^ 3} = = D2 y D3 e X, as it intersects only D- and D3. Proceeding like this, we can say that if £ = ^, then A(^) = {k e N: ||rjk — 0||C2 ^ ^} intersects (p — 1) of sets namely D-, D3, D4,... Dp and, hence, A( 1) e X. Now, by the Archimedean property, we can say that there exists 1 < £ that implies A(e) c A(-) e X. Hence, X — lim rjk = 0. Thus, our claim is established.
yp,
i'c-W
Theorem 1. Let (^k) be a sequence in C, such that £k-—^ C Then
£ is uniquely determined.
Proof. If possible, suppose: ^ -^ and -^ hold for
some £ ^ rj in C. Choose £ > 0, such that ||£ — rjHC2 = 2e. Then, by Definition 13, there exists M1, M- e T(X), such that the sequences (xk)
and (Vt) defined by x, - ' " P M' and y„ - I&' k P M- satisfy
I £, k R M]_, \rj, k R M,
the following properties:
• @£> 0, {ke N: ||xk — f ||c2 ^ £} e K,
• @£ > 0, {ke N: || yk — rjHc2 ^ e} e K.
Then, @£ > 0, {k e N: ||(xk — Uk) — (C — v)HC2 ^ £} e K. Now, as the inclusion, M1 x M- c {k e N: H(xk — yk) — (£ — v)HC2 ^ £} holds, so, by hereditary of K, Mi x M- e K, which implies N\(Mi x M-) e T(K). Again, as M1, M- e T(X), so M1 x M- e T(X). Now, N\(M1 x M-) e T(K)
and M1 x M2 e T{X) implies N\{M1 x M2) x {M1 x M2) e T{X v K) i.e 0 e T{X v K), a contradiction. □
Theorem 2. Let {^) and {rqk) be two sequences in C2, such that Çk-—^ C and rqk-r] hold. Then
/m ^ lK-W-W C2 ^
(i) ik + Vk-- £ + v,
-K,_........
- II - 11 C
(ii) c^k-> where ce R.
|| ■ || C || ■ || C
Proof. (i) Suppose £k -^ £ and r]k -^ -q. Then there exists M#i, M2 e T(X), such that the sequences (xk) and (yk) defined by
Uk, ke M-1 (rik, ke M2,
xk = < and yk = < are /C-convergent to E and
k [C, k R Mi, (/7, kRM2,
rq, respectively.
In other words, @e > 0, we have Di(e) = {k e N: \\xk — £\\C2 ^ 2} e fc and D2(e) = {k e N: \\yk — rj\\c2 ^ f} e fc. Now, as the inclusion (N\Di) x (N\D2) C {k e N: \\xk + yk — E, — r]\\c2 < e} holds, we must have {k e N: \\xk + yk — £ — r]\\c2 ^ e} C Di y D2 e fc.
| | ■ 11 C
Proving xk + yk-^ i + V. (!)
Now we take M = Mi x M2 e T(X) and define the sequence (Sk) as
Sk = i ^k + ^k, , Then, by virtue of (i), we have Sk-—^ £ + n.
k + rj, kR M. k ^ 1
IK-|| ■ 11 C2
This implies that ^ + r]k-> £ + rq.
(ii) Proof of this part is easy and so is omitted. □
1*- || ■ 11 C
Theorem 3. Let (^) be any sequence in C2, such that £k -^
1K- | | ■ 11 C2
Then £k-> for any admissible ideal fc.
1 *- | | ■ 11 c2
Proof. Let ^ -> £. Then, by Definition 8, there exists a set
M = {mi < m2 < ... < mk < ...} e T(X), such that the subsequence (E>mk) is convergent to £. This implies that for any e > 0, the set A = {k e N: \\rqk — £\\c2 ^ z} contains finite number of elements, where
fi. i \ ' ' k , k e M,
the sequence (%) is given by % = j ^ ^ r m
Since fc is admissible, so the above set A e fc and the result follows. □
Remark 1. The converse of the theorem mentioned earlier may not be valid.
Proof. The result is illustrated by the following example. □
Example 3. Consider Example 2. The sequence () is X^-convergent to 0 for K = X. But (^) is not X*-convergent to 0.
Justification. If £k-^ 0 holds, then, by Definition 8, there exists
a set T = {t1 < fa < ... < tk < ...} e T(X), such that
for any e > 0, there exists N = Ne e N: ||£tk — 0||C2 < e for all k > N. (2)
Now, as T e T(X), so T = N\ B for some B e X. By definition of X, there
j
exists some j e N, such that B c [J Dp, which as a consequence implies
p-1
Dj+1 c T. But then we have, £tk = for infinitely many k's, which is a contradiction to (2).
Theorem 4. Assuming (^) is an arbitrary sequence in C^ if (^) converges with respect to K to £ e C^ then the sequence (^) is also X^-convergent to
Proof. Since ^ ——£, so,
for any e > 0, { k e N: ||— f||C2 ^ e} e K. (3)
Choose M = N from T(X). Define the sequence % as % = £k,k e M. Then, using (3), we get for any e > 0, { k e N: ||% — £||C2 ^ £} e K, i.e.,
xk-||-||c2
r]k is ^—convergent to £. Hence, ^-> £. □
Remark 2. The reverse of the previously mentioned theorem may not necessarily be valid.
Proof. The result is illustrated by the following example. □
p-1
Example 4. Define Q1 = q1N y {1} and Qp = qpN\{J Di, p > 1, where
8
qp is the pth prime number. Then N = U Qp is a decomposition of N.
P-1
Consider the ideal that consists of those subsets of N that intersect with
finitely many Qp's, and another ideal X$. Let (^k) be any sequence in C2
1 \ei + e2, k is prime,
defined by = <
le^e2, k is not prime.
Then, (^k) is X$x-convergent to 0 but not X- convergent to 0.
Justification. Let P be the set of all prime numbers. Then 5(P) = 0, so P e X8. Let M = N\P. Then M e T(Xs). Now the sequence
(rib) defined as rib = < 1 1 is the null sequence and, therefore,
( ) [0, k R M,
(r]k) is X-convergent to 0. Hence, the sequence (^k) is X^-convergent to 0. But (^k) is not X-convergent to 0. Since for each e > 0, the set A(e) = {k e N: ||- 0}c2 ^ e} R X.
Theorem 5. Let X and K be two ideals in N, such that X C fc. Consider
-rlC ii ii y || ||
llc2 J. m f
a sequence (^) in C2, such that £k-^ Then ^-^
iA
Proof. Let X C fc hold, and suppose £k -> £. So, by definition, there exists M e T(X), such that the sequence (rjk) defined as
rqk = , ^ p m is fc-convergent to £, which as a consequence implies
@£ > 0 {ke M: ||Ck - e|c2 ^ £} e fc. (4)
Thus, {ke N: || - dk ^ e} C {k e M: || - dk ^ e}v (N\M) e fc, by equation (4) and our assumption X C fc. Hence, ^k-^ £. H
Remark 3. If a sequence is X^-convergent, then it may not be X-conver-gent to the same limit.
8
n
p
Example 5. Let Dp = {2p_lk: k be an odd number}. Then N = [J D
p-1
is a decomposition of N. Let X = X^, those subsets of N that intersect finitely many Dp's. Consider the sequence (^k) defined by ^k = 61_62, @k e Dp's. Then the sequence is X-convergent to 0. Let Xf consist of all finite subsets of N. Then £k is not Xxf-convergent to 0.
Theorem 6. Let X and fc be two ideals in N. Let (^k) be a sequence in
iK_||'||c2 2_||'||c2
C2. Then £k-y C implies ^-> £ if and only if fc C X.
1K- || ■ 11 C2
Proof. Let fc C X and suppose ^ -> £. Then the result follows
directly from the following inclusion:
{ k e N: \\Ck — ^€2 ^ £}c{ke M: \\& — £\\c2 > e} y (N\M).
For the converse part, let (I;k) be a sequence in C2, such that
1k- || ■ || C2 1- || ■ ||c2 I;k-> £ implies (;k-> £. To prove that, fc C X. Let us assume
the contrary. Then, there exists a set, say A e fc\X. Choose ^ e €2
such that ^ ^ £2.
'&, ke A,
fè:
Define a sequence {£k ) as £k = .
4 ) ^ ke{N\A).
Let e > 0 be arbitrary. Then, clearly, {k e N : || £k—£2||c2 ^ £} — A e K; this implies {^) is ^-convergent to £2. Thus, {^) is XK-convergent to £2. For e = ||^ — £2||c2 , {k e N : ||£k — £2||c2 ^ z} = A e X: a contradiction. Hence, we must have K — X. □
Theorem 7. Let X and K be two ideals in N, such that the condition
I- II ■ 11 c
AP{X, K,) holds. Then for a sequence {) in C2, -^ £ implies
û ^ e
x— - | ■ -1 c
Proof. Let -^ £. Choose a sequence of rationals (£i)iPN.
Then, for all i, Mi = {k e N : ||Çk — C||c2 < £i} e T{X). Then, by Definition 11, there exists a set M e T{X), such that for any i e N,
{M\Mi) e K. Consider the sequence (r]k) defined by r]k = i ^k, J
[t k R M.
To complete the proof, it is sufficient to show that the sequence (rqk) is ^-convergent to ^ e C2. Now,
{k e N : || ^ — £||c2 < = = {N\M) y {k e M : || ^ — £||c2 < £i} = {N\M) y {Mi x M) = N\{M\Mi ).
Now, as (M\Mi) e fc, so N\(M\Mi) e T(fc) and, consequently, we have
^_| | ■ | | ^
{ke N: \\rjk — f\\c2 < £i} e T(fc), i.e., rjk-^ f.
Hence, £k-^^ 6 □
Theorem 8. Let X, fc, Xi, X2, K,]_, fc2 be ideals in N and (^) be any sequence in C2. Then:
(i) If XKl — lim £k = l = XK2 — lim £k then XKlvK2 — lim £k = l.
(ii) If Xf — lim = l = — lim then (Xi v X2— lim = l.
Proof. (i) Since XKl — lim = l and X^2 — lim = l, so there exists M,N e T(X), such that for all e,8 > 0, {k e M: \\^ — l\\c2 > e} e Kx and {k e N: \\Ck — l\ \€2 ^ 5} e fc2.
By the hereditary property of fci and fc2, @£ > 0, 5 > 0, we have,
{ k e Mx N: \\& — l\c2 ^ 4 e fci, {k e Mx N: \\^ — l\\c2 ^ 5} e K2. (5)
Let r] > 0 be arbitrary. Then, from (5), choosing e = 8 = rj, we get {ke M x N: \\^ — l\\€2 > rj} e fci v fc2.
ke M x N,
k,
l,
As MxN e T(X), so the sequence (uk) defined by uk =
l, kR M x N,
is fci v fc2-convergent to l. Hence, XKlvlC2 — lim£k = l. (ii) Since Xf — lim £k = l and X^ — lim £k = l, so there exists M e T(Xi) and Ne T(X2), such that for all £,8 > 0, {ke M: \\— l||€2 ^ £} e fc and {k e N: \\& — l\\C2 ^ S} e fc.
By the Hereditary property of fc, we have for all e > 0, 8 > 0, {ke M x N: \\£k — l\\€2 ^ e} e fc and { k e M x N: \\— l\\€2 ^ ¿} e fc. Let r] > 0 be arbitrary. Choosing e = 8 = rj, we get,
@T] > 0, { k e M x N: \\— l\\€2 ^ r]} e fc.
As Mx N e T(Xi v X2), so we can conclude that (Xi v X2)K — lim £k = l. □
4. Conclusion. The main contribution of this paper is to provide the notion of XK-convergence of sequences of bi-complex numbers and study some of its properties and identify the relationships between newly introduced notion and its relationship with other convergence methods of sequences of bi-complex numbers. These ideas and results are expected to be a source for researchers in the area of convergence of sequences of bi-complex number. Also, these concepts can be generalized and applied for further studies.
References
[1] Bera S., Tripathy B. C. Statistical convergence in a bi-complex valued metric space. Ural Math. J. 2023, vol. 9, no. 1, pp. 49-63. DOI: https://doi .org/10.15826/umj.2023.1.004
[2] Bera S., Tripathy B. C. Statistical bounded sequences of bi-complex numbers. Probl. Anal. Issues Anal. 2023, vol. 12(30), no. 2, pp 3-16.
DOI: https://doi .org/10.15393/j3.art. 2023.13090
[3] Choudhury C., Debnath S. On X-convergence of sequences in gradual normed linear spaces. Facta. Univ. Ser. Math. Inform. 2021, vol. 36, no.3, pp. 595-604. DOI: https://doi.org/10.22190/FUMI210108044C
[4] Das P., Sleziak M., Toma V. XK — Cauchy functions. Comp and Math with Appli. 2014, vol. 173, pp. 9-27.
DOI: https://doi.org/10.1016/j.topol.2014.05.008
[5] Debnath S., Choudhury C. On some properties of XK— convergence. Palest. J. Math. 2022, vol. 11, no.2, pp. 129-135.
[6] Debnath S., Rakshit D. On X—statistical convergence. Iran. J. Math. Sci. Inform. 2018, vol. 13, no.2, pp. 101-109.
DOI: https://doi .org/10.7508/ijmsi.2018.13.009
[7] Kostyrko P., Salat T., Wilczynski W. X—convergence. Real. Anal. Exch. 2000/2001, vol. 26, no. 2, pp. 669-686.
[8] Macaj M., Sleziak M. XK- Convergence. Real. Anal. Exch. 2010-2011, vol. 36, pp. 177-194.
[9] Mursaleen M., Debnath S., Rakshit D. X-statistical limit superior and X-statistical limit inferior. Filomat. 2017, vol. 31, no. 7, pp. 2103-2108. DOI: https://doi .org/10.2298/FIL1707103M
[10] Nabiev A., Pehlivan S., Gurdal M. On X- Cauchy sequences. Taiwanese J. Math. 2007, vol. 11, no. 2, pp. 569-576.
DOI: https://doi .org/10.11650/twjm/1500404709
[11] Price G. B. An Introduction to Multi-complex Spaces and Functions. Monographs and Text books in Pure and Applied Mathematics. Marcel Dekker. Inc. New York. 1991.
[12] Rochon D., Shapiro M. On algebraic properties of bi complex and hyperbolic numbers. Anal.Univ. Oradea, Fasc. Math. 2004,vol. 11, pp. 71-110.
[13] Savas E., Das P. A generalized statistical convergence via ideals. Appl. Math. Lett. 2011, vol. 24, no.6, pp. 826-830.
DOI: https://doi .org/10.1016/j.aml.2010.12.022
[14] Scorza D. G. Sulla rappresentazione delle funzioni di variabile bicomplessa totalmente derivabili. Ann. Mat. 1934, vol. 5, pp. 597-665.
[15] Segre C. Le rappresentation reali delle forme complesse e gli enti iperalge-brici. Math. Ann. 1892, vol. 40, pp. 413-467. (in Italian)
DOI: https://doi .org/10.1007/BF01443559)
[16] Spampinato N. Estensione nel campo bicomplesso di due teoremi, del levi-Civita e del severi, per le funxione olomorfe di due variabili bicomplesse I, II. Reale Accad. Naz. Lincei. 1935, vol. 22, no. 6, pp. 96-102.
Received May 04, 2024. In revised form, September 09 , 2024. Accepted September 16, 2024. Published online October 12, 2024.
Department of Mathematics
Tripura University (A Central University)
Suryamaninagar-799022, Agartala, India.
J. Hossain
E-mail: [email protected]
S. Debnath (Corresponding Author) E-mail: [email protected]
