On series solutions of and applications
© Shivakumar and Zhang; licensee Springer 2013
Received: 13 February 2013
Accepted: 17 February 2013
Published: 4 March 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.
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
Next, we establish the recursive relation between and .
for suitable values of k and q.
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 .
Clearly, all coefficients are just in terms of and , that is, and .
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.
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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