Numbers of real and complex zeros of .
In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. Finally, we investigate the zeros of the 3-variable Hermite polynomials by using computer.
- differential equations
- heat equation
- Hermite polynomials
- the 3-variable Hermite polynomials
- generating functions
- complex zeros
Many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. The special polynomials of two variables provided new means of analysis for the solution of a wide class of differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems.
In , the Hermite polynomials are given by the exponential generating function
We can also have the generating function by using Cauchy’s integral formula to write the Hermite polynomials as
with the contour encircling the origin. It follows that the Hermite polynomials also satisfy the recurrence relation
Further, the two variables Hermite Kampé de Fériet polynomials defined by the generating function (see )
are the solution of heat equation
We note that
The 3-variable Hermite polynomials are introduced .
The differential equation and he generating function for are given by
By (2), we get
By comparing the coefficients on both sides of (3), we have the following theorem.
Applying Eq. (2), we obtain
On equating the coefficients of the like power of in the above, we obtain the following theorem.
Also, the 3-variable Hermite polynomials satisfy the following relations
The following elementary properties of the 3-variable Hermite polynomials are readily derived form (2). We, therefore, choose to omit the details involved.
The 3-variable Hermite polynomials can be determined explicitly. A few of them are
Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [7, 8, 12, 16, 17, 18, 19]). In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. In addition, we investigate the zeros of the 3-variable Hermite polynomials using numerical methods. Using computer, a realistic study for the zeros of the 3-variable Hermite polynomials is very interesting. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of the 3-variable Hermite polynomials.
2. Differential equations associated with the 3-variable Hermite polynomials
In this section, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials.
Then, by (4), we have
Continuing this process, we can guess that
Differentiating (7) with respect to , we have
Hence we have
Now replacing by in (7), we find
In addition, by (7), we have
It is not difficult to show that
Thus, by (14), we also find
From (10), we note that
Note that, here the matrix is given by
Therefore, we obtain the following theorem.
From (4), we note that
By the Leibniz rule and the inverse relation, we have
If we take in (21), then we have the following corollary.
For the differential equation
has a solution
3. Distribution of zeros of the 3-variable Hermite polynomials
This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover new interesting pattern of the zeros of the 3-variable Hermite polynomials . By using computer, the 3-variable Hermite polynomials can be determined explicitly. We display the shapes of the 3-variable Hermite polynomials and investigate the zeros of the 3-variable Hermite polynomials . We investigate the beautiful zeros of the 3-variable Hermite polynomials by using a computer. We plot the zeros of the for and (Figure 2). In Figure 2
Stacks of zeros of the 3-variable Hermite polynomials for from a 3-D structure are presented (Figure 3). In Figure 4
|Degree||Real zeros||Complex zeros|
|Degree||Real zeros||Complex zeros|
|7||−7.1098, −2.1887, −0.36350|
|9||−9.4984, − 4.6021, − 1.1118|
|10||−10.637, − 5.7212, − 1.5785, −0.61919|
|11||−11.745, − 6.8105, − 2.8680|
|12||−12.824, − 7.8743, − 3.8894, − 0.99513|
The plot of real zeros of the 3-variable Hermite polynomials for structure are presented (Figure 5).
Stacks of zeros of for , forming a 3D structure are presented (Figure 6). In Figure 6
It is expected that , has reflection symmetry analytic complex functions (see Figures 2–7). We observe a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials for . We also hope to verify a remarkable regular structure of the complex roots of the 3-variable Hermite polynomials for (Tables 1 and 2). Next, we calculated an approximate solution satisfying . The results are given in Tables 3 and 4.
|5||−1.3404, 1.4745, 6.6661|
|7||0.31213, 4.3783, 9.5946|
|8||−1.2604, 1.5304, 5.7274, 10.959|
|9||2.8224, 7.0271, 12.270|
|10||0.44594, 4.0615, 8.2834, 13.535|
|11||−1.1740, 1.5825, 5.2667, 9.5013, 14.760|
|12||−1.4659, −0.87728, 2.7469, 6.4398, 10.685, 15.949|
The plot of real zeros of the 3-variable Hermite polynomials for structure are presented (Figure 7).
Finally, we consider the more general problems. How many zeros does have? We are not able to decide if has distinct solutions. We would also like to know the number of complex zeros of Since is the degree of the polynomial , the number of real zeros lying on the real line is then , where denotes complex zeros. See Tables 1 and 2 for tabulated values of and . The author has no doubt that investigations along these lines will lead to a new approach employing numerical method in the research field of the 3-variable Hermite polynomials which appear in mathematics and physics. The reader may refer to [2, 11, 13, 20] for the details.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1A2B4006092).