Determination of the Properties of (<em>p</em>, <em>q</em>)‐Sigmoid Polynomials and the Structure of Their Roots (2024)

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• Jung Yoog Kang*

  • Department of Mathematics Education, Silla University, Busan, Republic of Korea

*Address all correspondence to: jykang@silla.ac.kr

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 n∈C, the pq-number is defined by

np,q=pn−qnp−q,∣qp∣<1.E1

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. Let z be any complex numbers with ∣z∣<1. The two forms of pq-exponential functions are defined by

ep,qz=∑n=0∞pn2znnp,q!,ep−1,q−1z=∑n=0∞qn2znnp,q!.E2

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

ep,qzep−1,q−1−z=1.E3

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. Let n≥k. pq-Gauss Binomial coefficients are defined by

nkp,q=np,q!n−kp,q!kp,q!,E4

where np,q!=np,qn−1p,q⋯1p,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 the pq-derivative operator of any function f, also referred to as the Jackson derivative, as follows:

Dp,qfx=fpx−fqxp−qx,x≠0,E5

and Dp,qf0=f′0.

If tx=∑k=0nakxk then Dp,qtx=∑k=0n−1ak+1k+1p,qxk, since Dp,qzn=np,qzn−1. This equation is equivalent to the pq-difference equation in q with 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,qfx−fqxDp,qgxgpxgqx=gpxDp,qfx−fpxDp,qgxgpxgqx.E7

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

∫fxdp,qx=p−qx∑k=0∞qkpk+1fqkpk+1x.E8

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

∑n=0∞En,p,qxtnnp,q!=2ep,qt+1ep,qtx.E9

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:

∑n=0∞Snxtnn!=1e−t+1etx.E10

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=x1 as one of its acute angles:

tanθ=f′x1=fx1x1−x2,x2=x1−fx1f′x1.

We continue the calculation scheme by computing

x3=x2−fx2f′x2,

or, in more general terms,

xn+1=xn−fxnf′xn,n=1,2,3,⋯.

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 K 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, fxn and f′xn. 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 ∣g′x∣<1. Since

gx=x−fxf′x,g′x=1−f′xf′x−fxf′′xf′x2=fxf′′xf′x2.

Hence if

fxf′′xf′x2<1

on an interval about the root r, the method will converge for any initial value x1 in the interval. The condition is sufficient only, and requires the unusual continuity and existence of fx and its derivatives. Note that f′x must not zero. In addition, Newton’s method is quadratically convergent and we can apply this method to polynomials.

Let f:D→D be a complex function, with D as a subset of C. We define the iterated maps of the complex function as the following:

fr:z0→f(f(⋯(f︸rz0⋯))

The iterates of f are the functions f,f◦f,f◦f◦f,…, which are denoted f1,f2,f3,.… If z∈C, and then the orbit of z0 under f is the sequence <z0,fz0,ffz0,⋯>.

Definition 1.7. The orbit of the point z0∈C under the action of the function f is said to be bounded if there exists M∈R such that ∣fnz0∣<M for all n∈N. If the orbit is not bounded, it is said to be unbounded.

Definition 1.8. Let f:D→C be a transformation on a metric space. A point z0∈D such that fz0=z0 is called a fixed point of the transformation.

We know that the fixed point is divided as follows. Suppose that the complex function f is analytic in a region D of C, and f has a fixed point at z0∈D. Then z0 is said to be:

an attracting fixed point if ∣f′z0∣<1;

a repelling fixed point if ∣f′z0∣>1;

a neutral fixed point if ∣f′z0∣=1.

If z0 is an attracting fixed point of f, then there exists a neighborhood of A such that if b∈A the orbit b converges 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 z0 is a repelling periodic point of f, then there is a neighborhood of N such that if b∈N, there are points in the orbit of b which are not in N. In the case of polynomials of degree greater than 0 and some rational functions, ∞ is also called an attracting fixed point, as, for each such function, f, there exist R>0 such that if ∣z∣>R then fnz→∞ as 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.

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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 define pq-sigmoid polynomials as following:

∑n=0∞Sn,p,qxtnnp,q!=1ep,q−t+1ep,qtx.E11

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

∑n=0∞Sn,p,q0tnnp,q!=∑n=0∞Sn,p,qtnnp,q!=1ep,q−t+1,E12

so we can be called Sn,p,q is pq-sigmoid numbers. We note that ep,q0=pn2 because of the property for pq-exponential function. If p=1 in the Definition 2.1, then one holds

∑n=0∞Sn,1,qxtnn1,q!=∑n=0∞Sn,qxtnnq!=1eq−t+1eqtx,E13

where Sn,qx is q-sigmoid polynomials. Moreover, if p=1,q→1 in the generation function of pq-sigmoid polynomials, we have

limq→1∑n=0∞Sn,1,qxtnn1,q!=∑n=0∞Snxtnn!=1e−t+1etx,E14

where Snx is sigmoid polynomials (see [16]).

Theorem 2.2. Let be ∣q/p∣<1. Then we get

ipn2xn=∑k=0nnkp,q−1n−kpn−k2Sk,p,qx+Sn,p,qx,ii∑k=0nnkp,q−1n−kpn−k2Sk,p,q+Sn,p,q=1ifn=0,0ifn≠0.E15

Proof. i Consider that ep,q−t≠−1. Then we can see

∑n=0∞Sn,p,qxtnnp,q!ep,q−t+1=ep,qtx.E16

Using power series of pq-exponential function in the equation above (16) and Cauchy product, we can compare both-sides as following:

∑n=0∞∑k=0nnkp,q−1n−kpn−k2Sk,p,qx+Sn,p,qxtnnp,q!=∑n=0∞pn2xntnnp,q!,E17

that is shown the required result of Theorem 2.2 i.

ii This equation is a recurrence formulae of pq-sigmoid numbers. We omit the proof of Theorem 2.2 ii since 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=1/4−p+q, S2,p,q=1/8p+qp2−3pq+q2, S3,p,q=−1/16p−qp+qp2−4pq+q2p2+pq+q2, S4,p,q=1/32p2+q2p8−4p7q−4p6q2+7p5q3+8p4q4+7p3q5−4p2q6−4pq7+q8, ⋯.

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

S0,p,qx=12S1,p,qx=141+2xS2,p,qx=18−p+q+2p+qx+4px2S3,p,qx=116q31+2x+2p2q−1+2x2+2pq2−1+2x2+p31−2x+4x2+8x3S4,p,qx=132−3p2q41+2x+q61+2x+4p3q3x−1+x+2x2+132p4q2−1+2x−3+4x2+pq5−3−2x+4x2+p5q3−2x+8x3+132p6−1+2x1−2x+4x2+8x3⋯. E18

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

iSn,p,qx=∑k=0nnkp,qpn−k2Sk,p,qxn−k,iiSn,p,qxy=∑k=0nnkp,qpn−k2Sk,p,qyn−k.E19

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

∑n=0∞Sn,p,qxtnnp,q!=∑n=0∞Sn,p,qtnnp,q!∑n=0∞pn2xntnnp,q!=∑n=0∞∑k=0nnkp,qpn−k2Sk,p,qxn−ktnnp,q!.E20

Therefore, we complete the proof of the Theorem 2.4 i at once.

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

∑n=0∞Sn,p,qxytnnp,q!=1ep,q−t+1ep,qtxep,qty=∑n=0∞∑k=0nnkp,qpn−k2Sk,p,qxyn−ktnnp,q!,E21

where the required result ii is completed immediately.□

Theorem 2.5. Let ∣q/p∣<1 and k be a nonnegative integer. Then we have

iSn,p,qx=∑k=0nnkp,q−1kpn−k2+k2qk2Sn,p−1,q−1−1xn−k,iiSn,p,q=−1npn2qn2Sn,p−1,q−1−1.E22

Proof. i Multiplying ep−1,q−1t in generating function of pq-sigmoid polynomials, we can investigate

∑n=0∞Sn,p,qxtnnp,q!=11+ep−1,q−1tep−1q−1tep,qtx=∑n=0∞Sn,p−1,q−1−1−tnnp−1,q−1!∑n=0∞pn2xntnnp,q!=∑n=0∞∑k=0nnkp,q−1kpn−k2+k2qk2Sn,p−1,q−1−1xn−ktnnp,q!.E23

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

ii Using 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

∑k=0nnkp,qpn−k2Sk,p,q=∑k=0nnkp,q−1kpn−k2+k2qk2Sn,p−1,q−1−1.E24

Theorem 2.7. For ∣q/p∣<1, pq-derivative of pq-sigmoid polynomials is as the follows:

Dp,qDp,qxSn,p,qx=np,qSn−1,p,qpx.E25

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

∑n=0∞Dp,qDp,qxSn,p,qxtnnp,q!=1ep,q−t+1Dp,qep,qtx=∑n=0∞Sn,p,qtnnp,q!∑n=0∞pn2Dp,qxntnnp,q!.E26

Here, we can note that Dp,qxn=pxn−qxnp−qx=np,qxn−1 (see [7]). From the equation above (26), we can transform the equation to

∑n=0∞Dp,qDp,qxSn,p,qxtnnp,q!=∑n=0∞∑k=0nnkp,qn−kp,qpn−k2Sk,p,qxn−1−ktnnp,q!=∑n=0∞∑k=0n−1np,qn−1kp,qpn−1−k2Sk,p,qpxn−1−ktnnp,q!.E27

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

Dp,qDp,qxSn,p,qx=np,q∑k=0n−1n−1kp,qpn−1−k2Sk,p,qpxn−1−k.E28

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

Theorem 2.8. Let ∣q/p∣<1 and p≠q. Then we investigate

Dp,qSn,p,qx=Sn,p,qpx−Sn,p,qqxp−qx.E29

Proof. Applying pq-derivative in the pq-exponential function, we have

Dp,q∑n=0∞Sn,p,qxtnnp,q!=∑n=0∞Dp,qSn,p,qxtnnp,q!=1ep,q−t+1ep,qptx−ep,qqtxp−qx=1p−qx∑n=0∞Sn,p,qpx−Sn,p,qqxtnnp,q!,E30

which is the required result. □

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

np,qp−qxSn−1,p,qpx+Sn,p,qqx=Sn,p,qpx.E31

Corollary 2.10. Putting x=1 in the Theorem 2.7 and Theorem 2.8, one holds

np,qp−q∑k=0n−1n−1kp,qpn−1−k2+n−1−kSk,p,q=Sn,p,qp−Sn,p,qq.E32

Theorem 2.11. Let ∣q/p∣<1, p≠q, and p,q≠0. Then we obtain

ppqxDSn,p,qx+Dp,qSn,p,qqx=qDp,qSn,p,qx+pqxDp,q2Sn,p,qx.E33

Proof. Using the Theorem 2.8 above, we have

∑n=0∞Dp,q2Sn,p,qxtnnp,q!=1p−qep,q−t+1ep,qptx−ep,qqtxx.E34

Here, we can obtain that

iDp,qep,qptxx=1pqx2qxDp,qep,qptx−ep,qpqtxDp,qx=qep,qp2tx−pep,qpqtxpqp−qx2,E35

and

iiDp,qep,qqtxx=1pqx2qxDp,qep,qqtx−ep,qq2txDp,qx=qep,qpqtx−pep,qq2txpqp−qx2.E36

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

∑n=0∞Dp,q2Sn,p,qxtnnp,q!=1p−qpqx∑n=0∞qDp,qSn,p,qpx−pDp,qSn,p,qqxtnnp,q!.E37

Therefore, we can see that

p−qpqxDp,q2Sn,p,qx=qDp,qSn,p,qpx−pDp,qSn,p,qqx,E38

and this shows the required result at once. □

Corollary 2.12. From the Theorem 2.11, one holds

p2qxDp,q2Sn,p,qx+pnp,qSn−1,p,qpqx=qnp,qSn−1,p,qp2x+pq2xDp,q2Sn,p,qx.E39

Theorem 2.13. Let a,b be any positive integers. Then we find

∑k=0nnkp,qSn−k,p,qaxSk,p,qbyan−kbk=∑k=0nnkp,qSn−k,p,qbxSk,p,qayakbn−k.E40

Proof. Suppose the form A is as the following.

A≔ep,qtxep,qtyep,q−ta+1ep,q−tb+1.E41

The form A of the equation above (41) can be transformed as

A≔∑n=0∞Sn,p,qaxtnannp,q!∑n=0∞Sn,p,qbytnbnnp,q!=∑n=0∞∑k=0nnkp,qSn−k,p,qaxSk,p,qbyan−kbktnnp,q!,E42

or, equivalently,

A≔∑n=0∞∑k=0nnkp,qSn−k,p,qbxSk,p,qaybn−kaktnnp,q!.E43

Comparing the coefficients of tn in both sides, we find the required result. □

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

∑k=0nnkp,qSn−k,p,qxSk,p,qbybk=∑k=0nnkp,qSn−k,p,qbxSk,p,qybn−k.E44

Corollary 2.15. When p=1 and q→1 in the Theorem 2.13, one holds

∑k=0nnkSn−kaxSkbyan−kbk=∑k=0nnkSn−kbxlSkayakbn−k,E45

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

Theorem 2.16. Let a,b be any integers without 0. Then we obtain

∑k=0nnkp,q−1nSn−k,p,qaxEk,p,qbyan−kbk=∑k=0nnkp,qSn−k,p,q−bxEk,p,q−ayakbn−k,E46

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

Proof. We consider the form B as

B≔ep,qtxep,qtyep,q−ta+1ep,qtb+1.E47

The form B of the equation above (47) can be transformed as

B≔12p,q∑n=0∞Sn,p,qaxtnannp,q!∑n=0∞En,p,qbytnbnnp,q!=12p,q∑n=0∞∑k=0nnkp,qSn−k,p,qaxEk,p,qbyan−kbktnnp,q!.E48

Also, we can transform the form B such as

B≔12p,q∑n=0∞∑k=0nnkp,qSn−k,p,q−bxEk,p,q−ay−1nbn−kaktnnp,q!.E49

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

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

∑k=0nnkp,q−1nSn−k,p,qxEk,p,qbybk=∑k=0nnkp,qSn−k,p,q−bxEk,p,q−ybn−k,E50

where En,p,qx is the pq-Euler polynomials.

Theorem 2.18. Let a,b be any integers without 0. Then we have

∑k=0nnkp,qSn−k,p,q−axEk,p,qbyan−kbk=∑k=0nnkp,qSn−k,p,q−bxEk,p,qayakbn−k,E51

where En,p,qx is the pq-Euler polynomials.

Proof. We set the form C as

C≔ep,qtxep,qtyep,qta+1ep,qtb+1.E52

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

C≔12p,q∑n=0∞Sn,p,q−axtn−annp,q!∑n=0∞En,p,qbytnbnnp,q!=12p,q∑n=0∞∑k=0nnkp,qSn−k,p,q−axEk,p,qby−an−kbktnnp,q!,E53

and

C≔12p,q∑n=0∞∑k=0nnkp,qSn−k,p,q−bxEk,p,qay−bn−kaktnnp,q!.E54

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

Corollary 2.19. Putting a=−1 in the Theorem 2.18, one holds

∑k=0nnkp,qSn−k,p,qxEk,p,qbybk=∑k=0nnkp,qSn−k,p,q−bxEk,p,q−y−1nbn−k,E55