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On series solutions of and applications
Advances in Difference Equations volume 2013, Article number: 47 (2013)
In this paper, we develop explicit expressions for the Taylor series coefficients in the formal Taylor series solution of the second-order linear differential equation for a given arbitrary function in terms of initial conditions. As applications, we apply our results to , and Airy’s equation and give explicit formulas for the Taylor coefficients of , which is a long-standing question.
The classical second-order linear differential equation
has been the subject of an innumerable amount of papers and of many classical mathematicians. In particular, a challenging problem has been the behavior of as x tends to infinity. Since no general formula solutions have been established before, one had to use certain indirect ways including numerical approaches (see, for example, [1, 2] and ).
In this paper, we present a constructive approach which yields explicit expressions for Taylor series solutions in terms of initial conditions. To the best of our knowledge, this is the first explicit formula giving solutions for differential equations of the type (1.1). In literature, no arbitrary method exists which gives complete explicit solutions for any for which there is a Taylor series solution as in the case of an ordinary point. It can easily be extended to regular singular points as well. It is well known that recurrence relations by their very nature cannot give analytic expressions for all the coefficients. When finding analytical solutions of differential equations of the type (1.1), for example, Airy’s equation, one might express solutions in some forms of Bessel functions or gamma functions, which often does not give much insight into the actual behavior of the solutions. Our results provide direct approaches for some problems. Also, our method can be used for evaluating precisely and very simply up to any number of terms of the series as well as for establishing convergence depending on and its derivatives. This method is new and novel and has a number of unexpected applications including the relationship between the zeros of a Taylor series and the Taylor coefficients, which is not included to keep the length of the paper to the permitted maximum length.
In the final part of this paper, we use our formulas to give one solution for a long-standing question: finding a Taylor series expansion for . Furthermore, this formula can be easily implemented in mathematical software like Maple  which provides more efficient ways to discuss these kinds of questions including parameter calculations.
2 Taylor series solutions
First we use (1.1) to give an expression of the n th derivative in terms of . For a suitable positive integer k, we have
Using induction (we omit the detailed proof due to the page limit), we obtain
Next, we establish the recursive relation between and .
for suitable values of k and q.
To establish this result, we will repeatedly make use of the following formula for interchanging finite sums:
From (2.3) with replaced by respectively, we get for the right-hand side (R.S.) of Lemma 2.1 (for simplicity consider ) as given by
Now, using (2.4) for the first two items above, we get
Again, using (2.4) for the second and third sums above, we get
From the above discussion and Lemma 2.1, the formal Taylor series solution at for can be written as follows.
In the first part of this section, we discuss three typical examples using our formulas, and in the second part, we give one solution for a long-standing question.
Example 3.1 Consider with , . From (2.2), we have . Now, using Theorem 2.2 and the results in Section 2 of , we obtain that tends to zero as .
Example 3.2 Consider the initial value problem , , . It is easy to get the general Taylor series solution by our formula as follows:
Example 3.3 It is well known that the general solution for Airy’s equation is given by , where and are linearly independent solutions of the Bessel equation of order . When we consider the general recurrence relation at , we have , . It is well known that although we can determine as many coefficients in terms of and , there is no known ‘formula’ for (see, for example, , pp.246-247). From our theory, the solution at can be written as follows:
Clearly, all coefficients are just in terms of and , that is, and .
Moreover, using our recursive formula, we can give the general term for the initial conditions , . By (2.1), it is easy to get
Using the initial conditions, the only non-zero term on the right-hand side of is when , that is, . Then we have the following recursive relation:
Note that . All terms with are zero. Combining (3.1) and (3.2) in , there is only one non-zero term in or . Furthermore, the coefficient of in is non-zero if and only if , and then it is easy to get that . Hence
Therefore, the radius of convergence equals ∞.
In the rest of this section, we describe a long-standing question: finding a Taylor series expansion for given by . For the history of this problem, see, e.g., Pounaltmadi . In  a recursive formula for in terms of is given by , with . There are no methods of finding explicitly for all k, although one can find it by actual computation for a finite number of ’s. This process is tedious and difficult, and also not feasible for large k. Now we can use our formulas to solve it.
Denote and . Then
where is given by
For at , and . The general Taylor series solution is given by
where all coefficients of x in the power series are in terms of ’s and hence in terms of ’s. In particular, if is a polynomial of degree m and is a polynomial of degree yielding , .
Our method can also be applied to other forms of functions. For example, consider . Then . Let . Applying above formula (3.3) gives a Taylor series expansion .
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The authors declare that they have no competing interests.
Each of the authors, YZ and PNS, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Shivakumar, P.N., Zhang, Y. On series solutions of and applications. Adv Differ Equ 2013, 47 (2013). https://doi.org/10.1186/1687-1847-2013-47
- Taylor series solutions
- the second-order ODE