Approximate zeros of

## Abstract

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.

### Keywords

- (p
- q)-sigmoid numbers
- (p
- q)-sigmoid polynomials
- (p
- q)-Euler polynomials
- roots structure
- fixed point
- 2000 Mathematics Subject Classification: 11B68
- 11B75
- 12D10

## 1. Introduction

In 1991, Chakrabarti and Jagannathan [1] introduced the

For any

Thereby, several physical and mathematical problems lead to the necessity of

**Definition 1.1.***Let**be any complex numbers with**. The two forms of**-exponential functions are defined by*

The useful relation of two forms of

In [9], Corcino created the theorem of

**Definition 1.2.***Let**-Gauss Binomial coefficients are defined by*

where

In 2013, Sadjang [21] derived some properties of the

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

and

If

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

Let

In 2016, Araci et al. [6] introduced a new class of Bernoulli, Euler and Genocchi polynomials based on the theory of

**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 [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

The calculation scheme follows immediately from the right triangle, which has the angle of inclination of the tangent line to the curve at

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

Hence if

on an interval about the root

Let

The iterates of

**Definition 1.7.***The orbit of the point**under the action of the function**is said to be bounded if there exists**such that**for all**. If the orbit is not bounded, it is said to be unbounded.*

**Definition 1.8.***Let**be a transformation on a metric space. A point**such that**is called a fixed point of the transformation.*

*We know that the fixed point is divided as follows. Suppose that the complex function**is analytic in a region**of**, and**has a fixed point at**. Then**is said to be:*

*an attracting fixed point if*

*a repelling fixed point if*

*a neutral fixed point if*

If

Based on the above, the contents of the paper are as follows. Section 2 checks the properties of

## 2. Some properties and identities of p q -sigmoid polynomials

This section introduces about

**Definition 2.1.***We define**-sigmoid polynomials as following:*

In the Definition 2.1, if

so we can be called

where

where

**Theorem 2.2.***Let be**. Then we get*

*Proof.*

Using power series of

that is shown the required result of Theorem 2.2

Based on the results from Definition 2.1 and Theorem 2.2, can be taken a few

**Example 2.3.***Some of the**-sigmoid polynomials are:*

**Theorem 2.4.***Let**be a nonnegative integer. Then we obtain*

*Proof.*

Therefore, we complete the proof of the Theorem 2.4

where the required result

**Theorem 2.5.***Let**and**be a nonnegative integer. Then we have*

*Proof.*

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

**Corollary 2.6.***From Theorem 2.5, one holds*

**Theorem 2.7.***For**-derivative of**-sigmoid polynomials is as the follows:*

*Proof.* Using

Here, we can note that

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

Applying the Theorem 2.4

**Theorem 2.8.***Let**and**. Then we investigate*

*Proof.* Applying

which is the required result.

**Corollary 2.9.***Comparing the Theorem 2.7 and Theorem 2.8, one holds*

**Corollary 2.10.***Putting**in 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

and

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.***Let**be 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

**Corollary 2.14.***Setting**in the Theorem 2.13, one holds*

**Corollary 2.15.***When**and**in the Theorem 2.13, one holds*

*where**is the sigmoid polynomials (see [*16*])*.

**Theorem 2.16.***Let**be any integers without**. Then we obtain*

where

*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.***Setting**in the Theorem 2.16, one holds*

*where**is the**-Euler polynomials*.

**Theorem 2.18.***Let**be any integers without**. Then we have*

*where**is the**-Euler polynomials*.

*Proof.* We set the form C as

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

and

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

**Corollary 2.19.***Putting**in the Theorem 2.18, one holds*

*where**is the**-Euler polynomials*.

## 3. Structure and various phenomena of roots of S n , p , q using the computer

This section mentions the structure of roots of

First, let us find an approximation of the root of

−1 | −1 | −0.99999 |

−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

**Conjecture 3.1.***Roots of**when**and**, always have two real roots.*

In the same way, we can find an approximation of the roots when the values of

Table 1 can be illustrated as Figure 1. The figure on the left is when

The following Figure 2 shows the structure of the roots when

Three-dimensional identification of Figure 2 shows the following Figure 3. Given the speculation, the roots are piling up near a point (

Based on the content above, we will now look at fixed points of

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

Let us look at the following by observing an application of an iterated

The structure of the roots of

## 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

## Acknowledgments

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).