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Determination of the Properties of (p, q)‐Sigmoid Polynomials and the Structure of Their Roots

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Jung Yoog Kang

Submitted: December 18th, 2019 Reviewed: February 22nd, 2020 Published: May 11th, 2020

DOI: 10.5772/intechopen.91862

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Nowadays, many mathematicians have great concern about p q -numbers, which are various applications, and have studied these numbers in many different research areas. We know that p q -numbers are different to q -numbers because of the symmetric property. We find the addition theorem, recurrence formula, and p q -derivative about sigmoid polynomials including p q -numbers. Also, we derive the relevant symmetric relations between p q -sigmoid polynomials and p q -Euler polynomials. Moreover, we observe the structures of appreciative roots and fixed points about p q -sigmoid polynomials. By using the fixed points of p q -sigmoid polynomials and Newton’s algorithm, we show self-similarity and conjectures about p q -sigmoid polynomials.


  • (p
  • q)-sigmoid numbers
  • (p
  • q)-sigmoid polynomials
  • (p
  • q)-Euler polynomials
  • roots structure
  • fixed point

1. Introduction

In 1991, Chakrabarti and Jagannathan [1] introduced the pq-number in order to unify varied forms of q-oscillator algebras in physics literature. Around the same time, Brodimas et al. and Arik et al. independently discovered the pq-number (see [2, 3]). Contemporarily, Wachs and White [4] introduced the pq-number in mathematics literature by certain combinatorial problems without any connection to the quantum group related to mathematics and physics literature.

For any nC, the pq-number is defined by


Thereby, several physical and mathematical problems lead to the necessity of pq-calculus. Based on the aforementioned papers, many mathematicians and physicists have developed the pq-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.Letzbe any complex numbers withz<1. The two forms ofpq-exponential functions are defined by


The useful relation of two forms of pq-exponential functions is taken by


In [9], Corcino created the theorem of pq-extension of binomials coefficients and found various properties which are related to horizontal function, triangular function, and vertical function.

Definition 1.2.Letnk.pq-Gauss Binomial coefficients are defined by


where np,q!=np,qn1p,q1p,q.

In 2013, Sadjang [21] derived some properties of the pq-derivative, pq-integration and investigated two pq-Taylor formulas for polynomials.

Definition 1.3.We define thepq-derivative operator of any functionf, also referred to as the Jackson derivative, as follows:


and Dp,qf0=f0.

If tx=k=0nakxkthen Dp,qtx=k=0n1ak+1k+1p,qxk, since Dp,qzn=np,qzn1. This equation is equivalent to the pq-difference equation in qwith known f, Dp,qgx=fx..

Theorem 1.4.This operator,Dp,q, has the following basic properties:

iDerivative ofaproductDp,qfxgx=fpxDp,qgx+gqxDp,qfx=gpxDp,qfx+fqxDp,qgx.E6
iiDerivative ofaratioDp,qfxgx=gqxDp,qfxfqxDp,qgxgpxgqx=gpxDp,qfxfpxDp,qgxgpxgqx.E7

Let fbe an arbitrary function. In [7], we note that the definition of pq- integral is


In 2016, Araci et al. [6] introduced a new class of Bernoulli, Euler and Genocchi polynomials based on the theory of pq-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 q-numbers are shown in a different shape by [17].

Definition 1.5.pq-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 [10] 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 x, we extrapolate along the tangent to its intersection with the x-axis, and take that as the next approximation. This is continued until either the successive x-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 x=x1as 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 Ktimes 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, fxnand fxn. We now use the result to show a criterion for convergence of Newton’s method. Consider the form xn+1=gxn. Successive iterations converge if gx<1. Since


Hence if


on an interval about the root r, the method will converge for any initial value x1in the interval. The condition is sufficient only, and requires the unusual continuity and existence of fxand its derivatives. Note that fxmust not zero. In addition, Newton’s method is quadratically convergent and we can apply this method to polynomials.

Let f:DDbe a complex function, with Das a subset of C. We define the iterated maps of the complex function as the following:


The iterates of fare the functions f,ff,fff,,which are denoted f1,f2,f3,.If zC, and then the orbit of z0under fis the sequence <z0,fz0,ffz0,>.

Definition 1.7.The orbit of the pointz0Cunder the action of the functionfis said to be bounded if there existsMRsuch thatfnz0<Mfor allnN. If the orbit is not bounded, it is said to be unbounded.

Definition 1.8.Letf:DCbe a transformation on a metric space. A pointz0Dsuch thatfz0=z0is called a fixed point of the transformation.

We know that the fixed point is divided as follows. Suppose that the complex functionfis analytic in a regionDofC, andfhas a fixed point atz0D. Thenz0is said to be:

an attracting fixed point iffz0<1;

a repelling fixed point iffz0>1;

a neutral fixed point iffz0=1.

If z0is an attracting fixed point of f, then there exists a neighborhood of Asuch that if bAthe orbit bconverges to z0. 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 z0is a repelling periodic point of f, then there is a neighborhood of Nsuch that if bN, there are points in the orbit of bwhich are not in N. In the case of polynomials of degree greater than 0and some rational functions, is also called an attracting fixed point, as, for each such function, f, there exist R>0such that if z>Rthen fnzas n.

Based on the above, the contents of the paper are as follows. Section 2 checks the properties of pq-sigmoid polynomials. For example, we look for addition theorem, recurrence relation, differential, etc. and find the properties associated with the symmetric property and pq-Euler polynomials. Section 3 identifies the structure and accumulation of roots of pq-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 pq-sigmoid polynomials to identify the domain leading to the fixed points.


2. Some properties and identities of pq-sigmoid polynomials

This section introduces about pq-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 pq-derivative, symmetric properties, and relations of pq-Euler polynomials for pq-sigmoid polynomials.

Definition 2.1.We definepq-sigmoid polynomials as following:


In the Definition 2.1, if x=0we can see that


so we can be called Sn,p,qis pq-sigmoid numbers. We note that ep,q0=pn2because of the property for pq-exponential function. If p=1in the Definition 2.1, then one holds


where Sn,qxis q-sigmoid polynomials. Moreover, if p=1,q1in the generation function of pq-sigmoid polynomials, we have


where Snxis sigmoid polynomials (see [16]).

Theorem 2.2.Let beq/p<1. Then we get


Proof.iConsider that ep,qt1. Then we can see


Using power series of pq-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 i.

iiThis equation is a recurrence formulae of pq-sigmoid numbers. We omit the proof of Theorem 2.2 iisince we can find the result for pq-sigmoid numbers to calculating the same method i.

Based on the results from Definition 2.1 and Theorem 2.2, can be taken a few pq-sigmoid numbers and polynomials can be calculated by using computer. We can observe that some of the pq-sigmoid numbers are S0,p,q=1/2, S1,p,q=1/4p+q, S2,p,q=1/8p+qp23pq+q2, S3,p,q=1/16pqp+qp24pq+q2p2+pq+q2, S4,p,q=1/32p2+q2p84p7q4p6q2+7p5q3+8p4q4+7p3q54p2q64pq7+q8, .

Example 2.3.Some of thepq-sigmoid polynomials are:


Theorem 2.4.Letkbe a nonnegative integer. Then we obtain


Proof.iUsing the definition of pq-exponential function, we can transform the Definition 2.1 as the follows:


Therefore, we complete the proof of the Theorem 2.4 iat once.

iiWe also consider pq-sigmoid polynomials in two parameters. Then we derive


where the required result iiis completed immediately.

Theorem 2.5.Letq/p<1andkbe a nonnegative integer. Then we have


Proof.iMultiplying ep1,q1tin generating function of pq-sigmoid polynomials, we can investigate


Comparing the both-side in the equation above, (23), we find the required results i.

iiUsing the same method i, we make the equation ii, so we omit the proof of Theorem 2.5 ii.

Corollary 2.6.From Theorem 2.5, one holds


Theorem 2.7.Forq/p<1,pq-derivative ofpq-sigmoid polynomials is as the follows:


Proof.Using pq-derivative for Sn,p,qx, we have


Here, we can note that Dp,qxn=pxnqxnpqx=np,qxn1(see [7]). From the equation above (26), we can transform the equation to


Using the comparison of coefficients in the both-sides, we can find


Applying the Theorem 2.4 iin the equation above, (28), we complete the proof of Theorem 2.7.

Theorem 2.8.Letq/p<1andpq. Then we investigate


Proof.Applying pq-derivative in the pq-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.Puttingx=1in the Theorem 2.7 and Theorem 2.8, one holds


Theorem 2.11.Letq/p<1,pq, andp,q0. Then we obtain


Proof.Using the Theorem 2.8 above, we have


Here, we can obtain that




Applying Eqs. (35) and (36) in the Eq. (34), we can catch the following equation:


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.Leta,bbe 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


or, equivalently,


Comparing the coefficients of tnin both sides, we find the required result.

Corollary 2.14.Settinga=1in the Theorem 2.13, one holds


Corollary 2.15.Whenp=1andq1in the Theorem 2.13, one holds


whereSnxis the sigmoid polynomials (see [16 ]).

Theorem 2.16.Leta,bbe any integers without0. Then we obtain


where En,p,qxis the pq-Euler polynomials (see [10]).

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


Comparing the equation above (48) and (49), we derive the result of Theorem 2.16. .

Corollary 2.17.Settinga=1in the Theorem 2.16, one holds


whereEn,p,qxis thepq-Euler polynomials.

Theorem 2.18.Leta,bbe any integers without0. Then we have


whereEn,p,qxis thepq-Euler polynomials.

Proof.We set the form C as


In the form C of the equation above (52), we can find that




Comparing the both sides in the equation above (53) and (54), we complete the required result of Theorem 2.18.

Corollary 2.19.Puttinga=1in the Theorem 2.18, one holds


whereEn,p,qxis thepq-Euler polynomials.


3. Structure and various phenomena of roots of Sn,p,qusing the computer

This section mentions the structure of roots of Sn,p,q. Furthermore, based on the previous content, an example of pq-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 pq-sigmoid polynomials. At this time, pand qshould not be the same value. If they have the same value, the denominator of pq-sigmoid polynomials will be 0. Consider the roots of S20,p,q. Once we fix the value of qto 0.1and switch pinto 0.9, 0.5, and 0.2, 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 − 0.400647i0.0334027 − 0.292274i0.083275 − 0.559033i
0.0110113 + 0.400647i0.0334027 + 0.292274i0.083275 + 0.559033i
0.132056 − 0.37596i0.119568 − 0.262599i0.136077
0.132056 + 0.37596i0.119568 + 0.262599i0.24766 − 0.490886i
0.239081 − 0.316515i0.189370.24766 + 0.490886i
0.239081 + 0.316515i0.191091 − 0.208427i0.379531 − 0.381018i
0.322803 − 0.228278i0.191091 + 0.208427i0.379531 + 0.381018i
0.322803 + 0.228278i0.241247 − 0.134952i0.471422 − 0.24088i
0.376059 − 0.119488i0.241247 + 0.134952i0.471422 + 0.24088i
0.376059 + 0.119488i0.265157 − 0.0476321i0.518528 − 0.0823508i
0.3943250.265157 + 0.0476321i0.518528 + 0.0823508i

Table 1.

Approximate zeros of S20,p,0.1x.

Here we can see that the approximate values of the roots change as the value of pchanges, as the roots always contain two real roots. Therefore, we can make the following assumptions.

Conjecture 3.1.Roots ofS20,p,qwhenp<1,n=20andq=0.1, always have two real roots.

In the same way, we can find an approximation of the roots when the values of qare changed in order of 0.9, 0.5, and 0.2and pis fixed at 0.1. In this case, when p=0.1and q=0.2, the approximate roots of S20,p,qall possess real roots which can be confirmed through Mathematica (x=395824,136973,47951,16837,5912,2076,728,255,89, 31,10,3,1,0,2,19,157,1231,9610,71387).

Table 1 can be illustrated as Figure 1 . The figure on the left is when p=0.9and the figure on the right is when p=0.2. 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 Sn,p,q. The structures of approximations roots in polynomials that combine the existing pq-number can be found becoming closer to a circle as the nincreases 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 nbecomes, it can be assumed that S20,p,qbecomes closer to the circle. Actually, a picture of S20,p,qcan be made using Mathematica.

Figure 1.

Approximate value of zeros forS20,p,0.1xforp=0.9,0.5,0.2.

The following Figure 2 shows the structure of the roots when nis 0to 50. When fixed at q=0.1, the figure on the left is when p=0.9and the right is when p=0.2.

Figure 2.

Zeros structures forSn,p,0.1xforp=0.9,0.5,0.2and0n50.

Three-dimensional identification of Figure 2 shows the following Figure 3 . Given the speculation, the roots are piling up near a point (x=1) and as the papproaches 1, the rest of the roots become closer to a circle as the value of nincreases.

Figure 3.

Zeros scattering forSn,p,0.1xforp=0.9,0.5,0.2and0n50.

Based on the content above, we will now look at fixed points of Sn,p,q. At this point, the value of qis fixed at 0.1and pis changed to 0.9, 0.5, and 0.2respectively. 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 pdecreases, it can be seen that the distance from the origin and the roots increases.

Figure 4.

Fixed points forS50,p,0.1xforp=0.9,0.5,0.2.

Similarly, the fixed points in the 3D structure can be checked as shown in Figure 5 .

Figure 5.

Scattering of fixed points forSn,p,0.1xforp=0.9,0.5and0n50.

Conjecture 3.2.Sn,p,qmay 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 S3,p,q, 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 p. For p=0.9and p=0.5, the number of the real roots of each of the five iterated function is 1,1,1,1,1,1, but for p=0.2, 3,3,3,3,3appears. The structure of the approximate value of the actual fixed point is shown in Figure 6 . The top part of Figure 6 is when p=0.9and the bottom figures represent p=0.2. Typically, most of the roots structures of general polynomials using q-numbers appear a circular shape, but it is difficult to find constant regularity in fixed points. However, sigmoid function including pq-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 pq-sigmoid polynomials will become a circular shape if nincreases tremendously.

Figure 6.

Scattering of fixed points forS3,p,0.1xiterated 5-times.

Let us look at the following by observing an application of an iterated S3,p,0.1using 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 pat 0.9and limit the range of values of xand yfrom 4to 4. Then the approximate root of S3,0.9,0.1becomes 0.936067, 0.1871690.256352i, 0.187169+0.256352i. Also, if the values going to 0.93606are shown in red, 0.1871690.256352iin blue, 0.187169+0.256352iin yellow, it is shown as the left of Figure 7 . The right side of Figure 7 is the picture that comes when S3,0.9,0.1are iterated twice.

Figure 7.

Classification of values that go near the fixed points ofS3,0.9,0.1.

The structure of the roots of Sn,p,qappears to have one value near 1and become a circular form as the nincreases. Also, as the value of pincreases, the diameter of the circle increases. A fixed point of Sn,p,qcan be seen to have a nearly constant form as pis reduced, which can also confirm an increase in the radius.


4. Conclusion

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 pq-number could be an useful data in deep learning using activation functions. Through iterating Sn,p,qwith 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).


Additional information

Mathematics Subject Classification: 11B68, 11B75, 12D10


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Written By

Jung Yoog Kang

Submitted: December 18th, 2019 Reviewed: February 22nd, 2020 Published: May 11th, 2020