Poly-Genocchi polynomials and its applications

: In this paper, we discussed some new properties on the newly deﬁned family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the ﬁrst time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional di ﬀ erential equation, including the delay fractional di ﬀ erential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is e ﬃ cient in solving various problems.

where Li k (x) = ∞ n=1 x n n k denotes the k th polylogarithm function. For x = 0, we obtain the poly-Genocchi numbers, g (k) n = G (k) n (0) of index k, where k is a positive integer. By using Eq (2.1), we obtain the following first few poly-Genocchi polynomials of index k, G (k) n (x).
The poly-Genocchi polynomials, G (k) n+1 (x) can be obtained if the lower degree n and lower index k are known. We now introduce the following theorem. Theorem 1. The poly-Genocchi polynomials, G (k) n (x) can be determined as follows: n (x), for n = odd, n (x), for n = even, where B n is Bernoulli number and G n is Genocchi number obtained using G n = 2(1 − 2 n )B n .
Proof. Suppose the generating function of poly-Genocchi polynomials as follows: Both sides of Eq (2.9) is then differentiated w.r.t t, which yields For the LHS of (2.10) and using (2.9), we denote B n as n th Bernoulli number, while G n as the Genocchi number, which yields (2.11) The first two terms in (2.11) are expanded, where after some algebraic manipulation, we obtain (2.12) By equating coefficients, when n is odd, we obtain On the other hand, for even n, we have (2.14) This completes the proof.

Orthogonal version of poly-Genocchi polynomials
Here, we briefly explain the Gram-Schmidt process for the poly-Genocchi polynomials. Note that we have G (k) 0 (x) = 0 and also G (k) 1 (x) = 1. Suppose also that φ 1 (x), · · · , φ q (x) are orthogonal version of poly-Genocchi polynomials obtained from Gram-Schmidt process in which the polynomial is orthogonal with respect to the inner product f, g = Obviously, we have φ 0 (x) = 0, φ 1 (x) = 1 and shifted Legendre polynomials, P 0 (x) = 1. Here, we compare the poly-Genocchi polynomials of degree q + 1 with the shifted Legendre polynomials of degree q since both of them have same highest power of x. By using Gram-Schmidt process as in (2.15) with q = 1, we obtain This is the same as degree 1 shifted Legendre polynomials, P 1 (x) = 2x − 1. In other words, the orthogonal version of poly-Genocchi polynomials, φ 2 (x), regardless of the k value in (2.1) or (2.2), is the multiple of 1 for degree 1 shifted Legendre polynomials after Gram-Schmidt process. Now, by using Gram-Schmidt process as in (2.15) with q = 2, we obtain After this Gram-Schmidt process, we obtain the orthogonal version of poly-Genocchi polynomials, φ 3 (x) in multiple of 2 for degree 2 shifted Legendre polynomials, P 2 (x). Upon continuing the Gram-Schmidt process as in (2.15) with q = 3 yields After this Gram-Schmidt process, we obtain the poly-Genocchi polynomials, φ 4 (x) in multiple of 5 for degree 3 shifted Legendre polynomials, P 3 (x). By using similar algebra manipulation, for φ 5 which is the multiple of 14 for degree 4 shifted Legendre polynomials, P 4 (x). We summarize the results as shown in Table 1. These multiples are indeed the Catalan numbers 1, 1, 2, 5, 14... which given by C n = 1 n+1 2n n . More generally, we have In conclusion, for the orthogonal version of poly-Genocchi polynomials, φ q (x) obtained via Gram-Schmidt process, we have q are the Catalan numbers, C(q). Table 1. Existing of Catalan numbers in the Gram-Schmidt process for poly-Genocchi polynomials, φ q (x).

Shifted poly-Genocchi polynomials sequence
This subsection extends the determinant form and recurrence relation of shifted Genocchi polynomials sequence recently introduced in [16] to shifted poly-Genocchi polynomials sequence. Similar to [16], we shift the order of poly-Genocchi polynomials from n to n + 1, i.e. we have n (x) denotes shifted poly-Genocchi polynomials sequence. We now have the following lemma.
Lemma 1. For n > 0, the determinant form and recurrence relation of shifted poly-Genocchi polynomials sequence, Gs (k) n (x) is given by 0 · · · · · · 0 s n−1,n−1 s n,n−1 (2.17) and The procedure to obtain the values for s i, j follows from Francesco A. Costabile et al. [16], summarized as follows: Step 1: From G (k) n (x) = n r=0 n r g (k) n−r x r , the poly-Genocchi number, g (k) i is obtained. Hence, we calculate the lower triangular Toeplitz matrix, T G with entries Step 2: The upper triangular, S , can be obtained via For example, the determinant form of shifted poly-Genocchi polynomials sequence, Gs (2) 3 (x) (k = 2, n = 3) and Gs (3) 4 (x) (k = 3, n = 4) are given by Gs .
It is easy to see that the above determinant forms give the poly-Genocchi polynomials as follows: (2.19)

Operational matrix based on poly-Genocchi polynomials
This section derives the new operational matrix based on poly-Genocchi polynomials and applies it to solve the fractional differential equations. This new operational matrix is called the generalization of the Genocchi operational matrix developed in [29].

1)
where P α is an N × N operational matrix of fractional derivative of order α in Caputo sense and is defined as follows: where θ n,r, j is given by θ n,r, j = n!g (k) Here g (k) n is the poly-Genocchi number and c j can be obtained from the inner product via (2.6).
Proof. From (2.4), we can write the poly-Genocchi polynomials in analytical form, where its fractional derivative is given as in (3.3).

Error bound
In this subsection, we briefly explain the error bound for the function approximation for arbitrary f (x) by using poly-Genocchi polynomials.
Let C T G (k) (x) be the best approximation of f (x) out of Y, we then have where M = max Proof. By using Taylor's series, we can write If we truncate the Taylor's series, the following bound may be obtained, Since C T G (k) (x) is the best approximation of f (x) out of Y and y 1 (t) ∈ Y, then from .
Taking the square root of both sides of (3.8) yields This proofs the error bound inequality as in (3.7).
In short, for each sub interval [x i , x i+1 ], i = 1, 2, · · · , n, f (x) has a local error bound of O(h

Collocation scheme based on poly-Genocchi operational matrix
In this subsection, we use the collocation scheme based on the poly-Genocchi operational matrix to numerically solve the fractional differential equation. This kind of approach replaces symbol by symbol, i.e. replacing fractional derivative, D α with the operational matrix, P α . This approach is also the same if we intend to solve the integer order differential equations, i.e. when α = 1. To do this, we have the following procedure: Step 1: We first approximate y(x) using poly-Genocchi polynomials as follows: where C = [c 1 , c 2 , · · · , c N ] is an unknown vector that need to be determined. If we want to approximate fractional derivative for y(x), we replace it by poly-Genocchi operational matrix as in For the initial and boundary conditions, we can replace y(0) = a with CG (k) (0) T − a = 0 and y(1 Step 2: Substituting (3.9) and (3.10) into the fractional differential equation, it collocates at the collocation points x i = i N , i = 1, 2, · · · , N − 2. Together with initial and boundary condition, we have N algebraic equations. We solve this system of algebraic equations with Newton's iterative method to obtain the value for C = [c 1 , c 2 , · · · , c N ] . Thus, the solution of fractional differential equation is obtained using (3.9).

Application in solving fractional differential equation
In this section, we solve some fractional differential equations to illustrate the applicability and accuracy of these poly-Genocchi polynomials. We achieve this by using the collocation scheme and the fractional derivative by employing an operational matrix based on poly-Genocchi polynomials. This operational matrix is the generalization of different indexes of Genocchi polynomials. All the numerical computations are carried out using Maple.
Example 1. Consider a simple fractional differential equation, given by with initial condition y(0) = 0. The exact solution is given by y(x) = x 2 .
This problem is solved using collocation scheme with N = 4, 8 and poly-Genocchi polynomials for k = 2 and k = 5. The absolute errors for the Example 1 are shown in Table 2. From the table, although the numerical scheme is simple and easy to use, the solution is accurate. Using different k values (i.e. k = 2, 5) of poly-Genocchi polynomials will give the same numerical result.
Example 2. Consider the following fractional delay differential equation as in [21,30].
with initial condition y(0) = 0, y (0) = 0, y (0) = 0 and h(x) = Γ(4) The exact solution is y(x) = x 3 . Here, we compare our results with those in [30] using the generalized Laguerre-Gauss collocation scheme with Laguerre parameters β and in [21] using a collocation scheme based on Genocchi operational matrix. Using N = 4 with poly-Genocchi polynomials k = 2 and k = 5, respectively, and following the procedure as in Example 2 [21], we obtained the result as in Table 3. Obviously, the proposed method with poly-Genocchi operational matrix gives better results.  Table 3. Comparison of the absolute errors obtained by the proposed method with those in [21,30] for Example 2.
[30] [21] Proposed Method Example 3. We consider the Lane-Emden equation up to the fractional order. It has been widely used in describing the thermal distribution profile in the human head [31] and radial stress on a rotationally symmetric shallow membrane cap [32]. The equation is given by: with initial and boundary conditions y (0) = 0, y(1) = 0.
The exact solution for α = 2 is given by y(x) = 2 ln B+1 Bx 2 +1 , where B = 3 − 2 √ 2. By using N = 8 with poly-Genocchi polynomials when k = 2, we obtain the approximate solution for α = 2, 1.9, 1.8 as in Figure 1. The numerical results are compared with the solution obtained in [33] using Modified Adomian Decomposition Method (MADM). It is obtained by combining between the Adomian Decomposition Method and collocation approach based on quintic B-spline basis function. The result in Table 4 shows that our proposed method with fewer terms is comparable with the result in [33]. Example 4. We consider the Bratu type equation as in [34]. Our proposed method are not only able to solve the integer order derivative for Bratu type equation, but also can solve the fractional Bratu type equation efficiently. This Bratu type equation are widely used in the fuel ignition model [35], the Chandrashekhar model [36]. Meanwhile, fractional Bratu type equations are studied for the problem arising in electro-spun organic nanofibers elaboration [37].
Here, we extend the Bratu type equation discuss in [34] to fractional order derivative as follow: with initial and boundary conditions y(0) = 0, y(1) = 0.

Conclusions
In this work, we investigated the new properties of poly-Genocchi polynomials, G (k) n (x), with any positive integer, k. When k = 1, it reduces to Genocchi polynomials. We successful derived the analytical expression to obtain a higher order and higher index of poly-Genocchi polynomials. We show that the orthogonal version of poly-Genocchi polynomials is the multiple of shifted Legendre polynomials. Interestingly, the multiple is given by Catalan numbers. We also extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence introduced in [16] to shifted poly-Genocchi polynomials sequence. We introduced the poly-Genocchi operational matrix for the first time, where the error bound for this new method is presented. Using a collocation scheme, we are able to solve the fractional differential equation and fractional delay differential equation. The numerical examples have shown that this proposed method is highly efficient and easy to use. Using few terms of poly-Genocchi polynomials in our proposed method can give more accurate results than existing methods. The method can be easily extended to solve more complicated problems such as those in [38,39].