Approximate zeros of .
Nowadays, many mathematicians have great concern about pq-numbers, which are various applications, and have studied these numbers in many different research areas. We know that pq-numbers are different to q-numbers because of the symmetric property. We find the addition theorem, recurrence formula, and pq-derivative about sigmoid polynomials including pq-numbers. Also, we derive the relevant symmetric relations between pq-sigmoid polynomials and pq-Euler polynomials. Moreover, we observe the structures of appreciative roots and fixed points about pq-sigmoid polynomials. By using the fixed points of pq-sigmoid polynomials and Newton’s algorithm, we show self-similarity and conjectures about pq-sigmoid polynomials.
- q)-sigmoid numbers
- q)-sigmoid polynomials
- q)-Euler polynomials
- roots structure
- fixed point
- 2000 Mathematics Subject Classification: 11B68
In 1991, Chakrabarti and Jagannathan  introduced the -number in order to unify varied forms of -oscillator algebras in physics literature. Around the same time, Brodimas et al. and Arik et al. independently discovered the -number (see [2, 3]). Contemporarily, Wachs and White  introduced the -number in mathematics literature by certain combinatorial problems without any connection to the quantum group related to mathematics and physics literature.
For any , the -number is defined by
Thereby, several physical and mathematical problems lead to the necessity of -calculus. Based on the aforementioned papers, many mathematicians and physicists have developed the -calculus in many different research areas (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]).
Definition 1.1.Letbe any complex numbers with. The two forms of-exponential functions are defined by
The useful relation of two forms of -exponential functions is taken by
In , Corcino created the theorem of -extension of binomials coefficients and found various properties which are related to horizontal function, triangular function, and vertical function.
Definition 1.2.Let.-Gauss Binomial coefficients are defined by
In 2013, Sadjang  derived some properties of the -derivative, -integration and investigated two -Taylor formulas for polynomials.
Definition 1.3.We define the-derivative operator of any function, also referred to as the Jackson derivative, as follows:
If then , since . This equation is equivalent to the -difference equation in with known , .
Theorem 1.4.This operator,, has the following basic properties:
Let be an arbitrary function. In , we note that the definition of - integral is
In 2016, Araci et al.  introduced a new class of Bernoulli, Euler and Genocchi polynomials based on the theory of -number and found some properties including difference equations, addition theorem, recurrence relations were derived. We observe some special properties and roots structures of Bernoulli, Euler, and tangent polynomials (see [4, 11, 16, 17, 18, 19, 20]). In particular, roots structures and fixed points of tangent polynomials including -numbers are shown in a different shape by .
Definition 1.5.-Euler polynomials are defined by
Several studies have investigated the sigmoid function for various applications (see [11, 12, 15, 16]). For example, a variant sigmoid function with three parameters has been employed to explain hybrid sigmoidal networks  and sigmoid function has been defined using flexible sigmoidal mixed models based on logistic family curves for medical applications [11, 12, 15].
Definition 1.6.We define the sigmoid polynomials as follows:
One of the most widely used methods of solving equations is Newton’s method. This method is also based on a linear approximation of the function, but does so using a tangent to the curve. Starting from an initial estimate that is not too far from a root , we extrapolate along the tangent to its intersection with the -axis, and take that as the next approximation. This is continued until either the successive -values are sufficiently close, or the value of the function is sufficiently near zero.
The calculation scheme follows immediately from the right triangle, which has the angle of inclination of the tangent line to the curve at as one of its acute angles:
We continue the calculation scheme by computing
or, in more general terms,
Newton’s algorithm is widely used because, at least in the near neighborhood of a root, it is more rapidly convergent than any of the methods so far discussed. The method is quadratically convergent, by which we mean that the error of each step approaches a constant times the square of the error of the previous step. The net result of this is that the number of decimal places of accuracy nearly doubles at each iteration. However, offsetting this is the need for two function evaluations at each step, and . We now use the result to show a criterion for convergence of Newton’s method. Consider the form . Successive iterations converge if . Since
on an interval about the root , the method will converge for any initial value in the interval. The condition is sufficient only, and requires the unusual continuity and existence of and its derivatives. Note that must not zero. In addition, Newton’s method is quadratically convergent and we can apply this method to polynomials.
Let be a complex function, with as a subset of . We define the iterated maps of the complex function as the following:
The iterates of are the functions which are denoted If , and then the orbit of under is the sequence
Definition 1.7.The orbit of the pointunder the action of the functionis said to be bounded if there existssuch thatfor all. If the orbit is not bounded, it is said to be unbounded.
Definition 1.8.Letbe a transformation on a metric space. A pointsuch thatis called a fixed point of the transformation.
We know that the fixed point is divided as follows. Suppose that the complex functionis analytic in a regionof, andhas a fixed point at. Thenis said to be:
an attracting fixed point if;
a repelling fixed point if;
a neutral fixed point if.
If is an attracting fixed point of , then there exists a neighborhood of such that if the orbit converges to . Attractive fixed points of a function have a basin of attraction, which may be disconnected. The component which contains the fixed point is called the immediate basin of attraction. If is a repelling periodic point of , then there is a neighborhood of such that if , there are points in the orbit of which are not in . In the case of polynomials of degree greater than and some rational functions, is also called an attracting fixed point, as, for each such function, , there exist such that if then as .
Based on the above, the contents of the paper are as follows. Section 2 checks the properties of -sigmoid polynomials. For example, we look for addition theorem, recurrence relation, differential, etc. and find the properties associated with the symmetric property and -Euler polynomials. Section 3 identifies the structure and accumulation of roots of -sigmoid polynomials based on the contents of Section 2 and checks the contents related to the fixed points. Also, we use Newton’s method to obtain a iterative function of -sigmoid polynomials to identify the domain leading to the fixed points.
2. Some properties and identities of -sigmoid polynomials
This section introduces about -sigmoid numbers and polynomials. From the generating function of these polynomials, we can observe some of the basic properties and identities of this polynomials. In particular, we can show the forms of -derivative, symmetric properties, and relations of -Euler polynomials for -sigmoid polynomials.
Definition 2.1.We define-sigmoid polynomials as following:
In the Definition 2.1, if we can see that
so we can be called is -sigmoid numbers. We note that because of the property for -exponential function. If in the Definition 2.1, then one holds
where is -sigmoid polynomials. Moreover, if in the generation function of -sigmoid polynomials, we have
where is sigmoid polynomials (see ).
Theorem 2.2.Let be. Then we get
Proof.Consider that . Then we can see
Using power series of -exponential function in the equation above (16) and Cauchy product, we can compare both-sides as following:
that is shown the required result of Theorem 2.2 .
This equation is a recurrence formulae of -sigmoid numbers. We omit the proof of Theorem 2.2 since we can find the result for -sigmoid numbers to calculating the same method .
Based on the results from Definition 2.1 and Theorem 2.2, can be taken a few -sigmoid numbers and polynomials can be calculated by using computer. We can observe that some of the -sigmoid numbers are , , , , , .
Example 2.3.Some of the-sigmoid polynomials are:
Theorem 2.4.Letbe a nonnegative integer. Then we obtain
Proof.Using the definition of -exponential function, we can transform the Definition 2.1 as the follows:
Therefore, we complete the proof of the Theorem 2.4 at once.
We also consider -sigmoid polynomials in two parameters. Then we derive
where the required result is completed immediately.
Theorem 2.5.Letandbe a nonnegative integer. Then we have
Proof.Multiplying in generating function of -sigmoid polynomials, we can investigate
Comparing the both-side in the equation above, (23), we find the required results .
Using the same method , we make the equation , so we omit the proof of Theorem 2.5 .
Corollary 2.6.From Theorem 2.5, one holds
Theorem 2.7.For,-derivative of-sigmoid polynomials is as the follows:
Proof. Using -derivative for , we have
Using the comparison of coefficients in the both-sides, we can find
Applying the Theorem 2.4 in the equation above, (28), we complete the proof of Theorem 2.7.
Theorem 2.8.Letand. Then we investigate
Proof. Applying -derivative in the -exponential function, we have
which is the required result.
Corollary 2.9.Comparing the Theorem 2.7 and Theorem 2.8, one holds
Corollary 2.10.Puttingin the Theorem 2.7 and Theorem 2.8, one holds
Theorem 2.11.Let,, and. Then we obtain
Proof. Using the Theorem 2.8 above, we have
Here, we can obtain that
Therefore, we can see that
and this shows the required result at once.
Corollary 2.12.From the Theorem 2.11, one holds
Theorem 2.13.Letbe any positive integers. Then we find
Proof. Suppose the form A is as the following.
The form A of the equation above (41) can be transformed as
Comparing the coefficients of in both sides, we find the required result.
Corollary 2.14.Settingin the Theorem 2.13, one holds
Corollary 2.15.Whenandin the Theorem 2.13, one holds
whereis the sigmoid polynomials (see ).
Theorem 2.16.Letbe any integers without. Then we obtain
where is the -Euler polynomials (see ).
Proof. We consider the form B as
The form B of the equation above (47) can be transformed as
Also, we can transform the form B such as
Corollary 2.17.Settingin the Theorem 2.16, one holds
whereis the-Euler polynomials.
Theorem 2.18.Letbe any integers without. Then we have
whereis the-Euler polynomials.
Proof. We set the form C as
In the form C of the equation above (52), we can find that
Corollary 2.19.Puttingin the Theorem 2.18, one holds
whereis the-Euler polynomials.
3. Structure and various phenomena of roots of using the computer
This section mentions the structure of roots of . Furthermore, based on the previous content, an example of -sigmoid polynomials is taken to identify the shape of the fixed points and the iterative function. And by applying it, we can find properties of self-similarity by using Newton’s method.
First, let us find an approximation of the root of -sigmoid polynomials. At this time, and should not be the same value. If they have the same value, the denominator of -sigmoid polynomials will be . Consider the roots of . Once we fix the value of to and switch into , , and , the following Table 1 can be found:
|−0.390901 − 0.133409i||−0.319498 − 0.0466997i||−0.632357 − 0.0806528i|
|−0.390901 + 0.133409i||−0.319498 + 0.0466997i||−0.632357 + 0.0806528i|
|−0.325772 − 0.249143i||−0.22888 − 0.212259i||−0.574469 − 0.251005i|
|−0.325772 + 0.249143i||−0.22888 + 0.212259i||−0.574469 + 0.251005i|
|−0.229438 − 0.33589i||−0.149885 − 0.266689i||−0.458705 − 0.409513i|
|−0.229438 + 0.33589i||−0.149885 + 0.266689i||−0.458705 + 0.409513i|
|−0.113312 − 0.387358i||−0.0593879 − 0.293992i||−0.102928 − 0.574072i|
|−0.113312 + 0.387358i||−0.0593879 + 0.293992i||−0.102928 + 0.574072i|
|0.0110113 − 0.400647i||0.0334027 − 0.292274i||0.083275 − 0.559033i|
|0.0110113 + 0.400647i||0.0334027 + 0.292274i||0.083275 + 0.559033i|
|0.132056 − 0.37596i||0.119568 − 0.262599i||0.136077|
|0.132056 + 0.37596i||0.119568 + 0.262599i||0.24766 − 0.490886i|
|0.239081 − 0.316515i||0.18937||0.24766 + 0.490886i|
|0.239081 + 0.316515i||0.191091 − 0.208427i||0.379531 − 0.381018i|
|0.322803 − 0.228278i||0.191091 + 0.208427i||0.379531 + 0.381018i|
|0.322803 + 0.228278i||0.241247 − 0.134952i||0.471422 − 0.24088i|
|0.376059 − 0.119488i||0.241247 + 0.134952i||0.471422 + 0.24088i|
|0.376059 + 0.119488i||0.265157 − 0.0476321i||0.518528 − 0.0823508i|
|0.394325||0.265157 + 0.0476321i||0.518528 + 0.0823508i|
Here we can see that the approximate values of the roots change as the value of changes, as the roots always contain two real roots. Therefore, we can make the following assumptions.
Conjecture 3.1.Roots ofwhen,and, always have two real roots.
In the same way, we can find an approximation of the roots when the values of are changed in order of , , and and is fixed at . In this case, when and , the approximate roots of all possess real roots which can be confirmed through Mathematica (, ).
Table 1 can be illustrated as Figure 1. The figure on the left is when and the figure on the right is when . Here we can see that the structure of the roots is changing. The following will identify the structure and the build-up of the roots of . The structures of approximations roots in polynomials that combine the existing -number can be found becoming closer to a circle as the increases and can be seen that a single root is continuously stacked at a certain point. Also, as the roots continue to pile up near at some point and the larger the becomes, it can be assumed that becomes closer to the circle. Actually, a picture of can be made using Mathematica.
The following Figure 2 shows the structure of the roots when is to . When fixed at , the figure on the left is when and the right is when .
Three-dimensional identification of Figure 2 shows the following Figure 3. Given the speculation, the roots are piling up near a point () and as the approaches 1, the rest of the roots become closer to a circle as the value of increases.
Based on the content above, we will now look at fixed points of . At this point, the value of is fixed at and is changed to , , and respectively. This can be found as shown in the following Figure 4. Figure 4 shows a nearly circular appearance and always has the origin. Also, as the value of decreases, it can be seen that the distance from the origin and the roots increases.
Similarly, the fixed points in the 3D structure can be checked as shown in Figure 5.
Conjecture 3.2.may have one fixed point which is the origin and the rest of the fixed points appear in the form of a circle.
The following is a third polynomial of , using a iterative function to find the approximate value of the fixed point. First, by iterating this third function five times and getting the number of real roots, the value of fixed points will vary depending on the value of . For and , the number of the real roots of each of the five iterated function is , but for , appears. The structure of the approximate value of the actual fixed point is shown in Figure 6. The top part of Figure 6 is when and the bottom figures represent . Typically, most of the roots structures of general polynomials using -numbers appear a circular shape, but it is difficult to find constant regularity in fixed points. However, sigmoid function including -numbers might have a special property for fixed points. In other words, we can guess from Figure 6 that the structure of fixed points for -sigmoid polynomials will become a circular shape if increases tremendously.
Let us look at the following by observing an application of an iterated using Figure 6. Let us try using the Newton’s method that we know well. Let us divide the values that go to the root of the tertiary function. First, fix at and limit the range of values of and from to . Then the approximate root of becomes , , . Also, if the values going to are shown in red, in blue, in yellow, it is shown as the left of Figure 7. The right side of Figure 7 is the picture that comes when are iterated twice.
The structure of the roots of appears to have one value near and become a circular form as the increases. Also, as the value of increases, the diameter of the circle increases. A fixed point of can be seen to have a nearly constant form as is reduced, which can also confirm an increase in the radius.
Sigmoid function is a very important function in deep learning. In the current situation of artificial intelligence development, the properties and speculations of the sigmoid polynomials revealed in this paper in the area of using -number could be an useful data in deep learning using activation functions. Through iterating with these properties, it can be assumed to have self-similarity and can be studied further to confirm new properties.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (No. 2017R1E1A1A03070483).