Instruction


FACTA UNIVERSITATIS   

Series: Electronics and Energetics Vol. 29, N
o
 4, December 2016, pp. 689 - 700 

DOI: 10.2298/FUEE1604689D 

A NOVEL ANALYTICAL METHOD FOR THE SELECTIVE 

MULTIPLIERLESS LINEAR-PHASE 2D FIR FILTER FUNCTION

 

Jelena R. Djordjević-Kozarov, Vlastimir D. Pavlović 

University of Niš, Faculty of Electronic Engineering, Niš, Serbia 

Abstract. In this paper, a novel analytical method for new class of selective linear-phase 

two-dimensional (2D) finite impulse response (FIR) filter functions generated by applying 

a new modified 2D Christoffel–Darboux formula for classical orthogonal Chebyshev 

polynomials of the first and the second kind is proposed. Fundamental research proposed 

in this paper is also illustrated by examples of 2D FIR filter and adequate comparison 

with new class of multiplierless linear-phase 2D FIR filter function given in the literature. 

Key words: 2D FIR filter function, Multiplierless, Linear-phase, Frequency response 

analysis, Chebyshev polynomials, Hilbert transform 

1. INTRODUCTION 

Successful applications of powerful orthogonal polynomials, in the filter theory, are 

well-known and described in [1]. A lot of problems in various scientific and technical 

areas have been solved applying the classical Christoffel-Darboux formula for all classic 

orthogonal polynomials. The new class of the explicit filter functions for continuous 

signals, generated by the classical Christoffel-Darboux formula for the classical Jacobi 

orthogonal polynomials, is given in detail in [2].  

Design of the linear-phase FIR filters for defined specifications is discussed in [3]. 

The grid density requirement for the design of FIR filters, with a useful design rule, is 

presented in detail in [4]. In [5] the authors present the relationship between the accuracy 

and the frequency grid density in 2-D filter designs. A new formula for determining the 

frequency grid spacing is proposed. One-dimensional half-band linear-phase FIR filter 

design approach is efficiently used in realization of 2D linear-phase FIR filter [6]. 

The paper [7] describes the approach for the successful design of 2D FIR filters with 

multipliers. Moreover, 2D FIR filters with nonstandard specifications are designed using 

transformation technique in [8, 9]. They are based on transformation of one-dimensional FIR 

filters, as well as direct application of the approximation techniques in two dimensions. 

                                                                 
Received October 13, 2015; received in revised form April 16, 2016 

Corresponding author: Jelena R. Djordjević-Kozarov 

University of Niš, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, Serbia 

(email: jelena.djordjevic-kozarov@elfak.ni.ac.rs) 



690 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

A simple recurrence formula for computing the impulse response coefficients of the 

sinc
N
FIR filter, consisting of a cascade of N sinc filters, each of length M, is presented in [11]. 

The initial consideration for the synthesis of the 2D FIR filter functions is given in a 

short paper [12]. Proposed Christoffel-Darboux formula for four orthogonal polynomials 

on two equal finite intervals for powerfully generating filter functions is proposed. In [13] 

is described in detail the analytical method for the synthesis of the multiplierless linear-

phase 1D and 2D FIR filter functions in an explicit form using Chebyshev orthogonal 

polynomials of the first kind. In [14] is described an analytical method for the synthesis of 

the multiplierless linear-phase 2D FIR filter functions in a compact form that can have the 

effect of Hilbert transformer in the z2 domain. A novel analytical method for a new class of 

linear-phase 2D FIR filter functions with a full effect of Hilbert transformer in z1 and z2 

domains is proposed in [15]. 

The main motivation for this research is the extreme property of Christofell-Darboux 

sum for the classical orthogonal polynomials that provides new results in the continuous 

domain, the domain of 1D z and 2D z domain. These results are written in explicit form and 

give a huge contribution to the filter theory. It should be emphasized that the multiplierless 

solutions are new solutions worthy of attention. There is no effect of the final quantization of 

filter coefficients because all the filter coefficients are equal per module. 

This paper presents further generalization of the previous research [4] in two dimensions. 

The proposed solution is a filter function in the z1 domain, and the Hilbert transformer in the 

z2 domain, and with the solution from [14] constitutes the whole. An analytical method of 

the Christoffel-Darboux formula for the classical orthogonal Chebyshev polynomials, of the 

first and the second kind, is proposed in this paper in an explicit form in continuous domain. 

Also, the new class of the linear-phase 2D FIR digital filters, generated by the proposed 

modified formula and by direct mapping from the continuous domain into 2D z domain, is 

given. In order to illustrate, the examples of the efficient design of the new class of selective 

linear-phase 2D FIR filter functions are also given. 

2. REVIEW OF THE 2D FIR FILTER FUNCTION 

Multiplierless linear-phase 2D FIR filter function with two free real parameters is 

considered in this paper. A linear-phase 2D FIR filter of (M x M)-order is defined by 

 1 2 1 1 2
0 0

( , , ) ( , , )
M M

r k

r k

H M z z K h M r k z z
 

 

    (1) 

where M is desired order of the filter, 
1

K  is the real constant and ( , , )h M r k  are the impulse 

response coefficients that are real numbers. Squared filter frequency response, in absolute units, 

can be presented by 

 1 21 2 1 2 1( , , ) ( , , ) , for , 2
i j

H M z z H M z z z e z e
 

   (2) 

or 

 
2

1 2 1 2 1 2( , , ) ( , , ) ( , , )
i j i j i j

H M e e M e e H M e e
      

  (3) 

Alternatively in dB, squared filter frequency response can be presented by 



 A Novel Analytical Method for the Selective Multiplierless Linear-phase 2D FIR Filter Function 691 

 1 2( ) 20 log ( , , )
i j

a dB H M e e
 

  (4) 

3. NEW CLASS OF TWO-DIMENSIONAL LINEAR PHASE MULTIPLIERLESS 

FIR FILTER FUNCTIONS  

Directly applying the formula proposed by Eq. (A.8), the new class of non-causal two-

dimensional symmetric FIR filter functions can be obtained as 

 

1 1 1 1 1 1

1 1 1 1 1 1

1 2

1 2 1
1 1 1 1 1 11

2 2 2 2 2 2

1 2

sin
12 2 2

( ) ( )
( , , )

sin
12 2 2

( ) ( )

M

r

z z z z z z
T U

r ri i

h r h r
H M z z K

z z z z z z
T U

r rj j

h r h r

     

     

        
      
      


 
   
        
            
  
 

 (5) 

or 

 
2 2 2 2

1 2 1 1 1 2 2
(M, , ) [( ) ( )]

1

M
r r r r

r

H z z ij K z z z z
   



      (6) 

Multiplying the Eq. (6) with factor 
2 2

1 2

N N
z z
 

 , the filter function can be generated as 

 
2 2 2 2 2 2 2 2

1 2 1 1 1 2 2
1

( , , ) [( ) ( )]
M

M r M r M r M r

r

H M z z ij K z z z z
       



      (7) 

It is obvious from Eq. (7) that the linear-phase FIR filter contains no multipliers and 

has only adders. 

The frequency response, 1 2( , , )
i j

H M e e
 

, can be defined as 

 1 2
( 2 ) ( 2 )

1 22 2
 1  2

0

( , , sin (2 ) sin (2 ))
i M j M M

i j

r

H M e e e e r r

 
 

 
 

   



    (8) 

and the magnitude characteristic is defined as 

 1 2
1 2

0

( , , ) sin (2 ) sin (2 )
Mi j

r

H M e e r r
 

 


   (9) 

and the amplitude characteristic, 
1 2

(2 , , )A M   , is defined as 

 
1 2 1 2

0

( , , ) sin (2 ) sin (2 )
M

r

A M r r   


   (10) 

The linear-phase function, 
1 2

(2 , , )M   , can be defined as 

 
( 2 ) ( 2 )

1 22 2
1 2

( , , )
i M j M

e eM

 
 

  
   

  (11) 



692 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

A filter function of K = 2R  cascaded identical blocks can be written as (H(M, z1, z2))
2R

. If 

we propose that the filter function H(M, 2R, z1, z2) is performed as a product of three 

functions of successive orders 1M , M  and 1M , than the form of the filter function 

can be given by: 

 
2

1 2 1 2 1 2 1 2
(2 , , , ) [ ( 1, , ) ( , , ) ( 1, , )]

R
H R M z z H M z z H M z z H M z z    (12) 

4. EXAMPLES OF THE NEW CLASS OF TWO-DIMENSIONAL LINEAR PHASE  

FIR FILTER FUNCTIONS 

The proposed design algorithm, for original 2D FIR filter function, has limitations in 

addition to the filter dimension (6MR x 6MR) and in addition to the value of two free real 

integer parameters 2R and M. This means that the linear-phase characteristic forms are 

limited. In Table 1, for some low values of 2R and M, the form of linear-phase 

characteristics and type of filter functions are given. When 2R is an even number, the 

proposed filter function has filter properties in both domains. 

Table 1 Explicit form of the linear phase characteristics of the proposed FIR filter for 

some low values of free integer parameters 2R and M 

2R M 1
( 4 )

2( 4 ) 2
2

 1  2
2( , , , )

j MR
i MR

M R e e


 



  
 

 

  
Type 

4 4 ( 32 ) ( 32 )1 2
 1  2

(4, 4, , )
i j

e e
   

  
   

  
z1 Filter 

z2 Filter 

4 5 40 401 2
 1  2( 5, 4, , )

i i j j
e e

   
  

   
   

z1 Filter 

z2 Filter 

4 6 
( 48 ) ( 48 )

1 2

 1  2
(6, 4, , )

i j

e e
   

  
   

  
z1 Filter 

z2 Filter 

4 7 
( 56 )( 56 )

1 2

 1  2
(7, 4, , )

ji

e e
  

  
  

  
z1 Filter 

z2 Filter 

4 8 
( 64 )( 64 )

1 2

 1  2
(8, 4, , )

ji

e e
  

  
  

  
z1 Filter 

z2 Filter 

4 9 
( 72 ) ( 72 )

1 2

 1  2
(9, 4, , )

i j

e e
   

  
   

  
z1 Filter 

z2 Filter 

4 10 
( 80 )( 80 )

1 2

 1  2
(10, 4, , )

ji

e e
  

  
  

  
z1 Filter 

z2 Filter 

4 11 
( 88 ) ( 88 )

1 2

 1  2
(11, 4, , )

i j

e e
   

  
   

  
z1 Filter 

z2 Filter 

Using the standard technique, the amplitude, magnitude and phase characteristics are 

obtained from Eq. (7) for the numerical values 2R = 4 and M = 6, and detailed 

characteristics of the filter function 
1 2

( , , ,2 )H M zR z  are given in the following figures. 



 A Novel Analytical Method for the Selective Multiplierless Linear-phase 2D FIR Filter Function 693 

 

(a)     (b) 

Fig. 1 a) 3D plot of normalized amplitude characteristics of proposed  

2D FIR filter for 2R =4 and M=6; b) Zoomed panel a) 

Illustrated examples of pass-band and stop-band characteristics of the considered FIR 

filter function for given values of attenuation, 
1 2

(2 , , , )a R M   , are shown in Fig. 2.  

  
(a)     (b) 

Fig. 2 2D contour plot of normalized magnitude characteristics: a) shape of the pass-band 

with attenuation of 0.28 dB for 2R =4 and M=6; b) shape of the stop-band with 

attenuation of 100 dB for 2R =4 and M=6 

Fig. 3 shows the phase characteristic of the considered linear-phase multiplierless 2D 

FIR filter function in the initial part, for the same values of the free integer parameters 2R 

and M, i.e., 2R = 4 and M = 6. 



694 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

 

Fig. 3 3-D plot of the phase characteristic of proposed 2D FIR filter for 2R = 4 and M = 6 

5. COMPARISON 

In this part of the paper, the comparison of the proposed solution for the multiplierless 

linear-phase 2D FIR filter functions and the solution described in [13] is discussed. For the 

same values of the real free parameters, M and 2R, and the same value of constant group 

delay we have considered the comparison of amplitude response characteristics and cut-off 

frequencies of the pass-band of filter and cut-off frequencies of the stop-band of filter. 

For the value of the free integer parameter 2R = 4, the paper described in [13] and this 

paper have  the same filter property both in z1 and in z2 domain. For the same odd M = 6 

we discussed the comparisons between the amplitude responce characteristics, as well as 

cut-off frequencies of the pass-band of filter with defined attenuation of 0,28 dB and cut-

off frequencies of the stop-band of filter with attenuation of 100 dB. 

We correctly compare two examples of filter functions that have the same values of 

free real integer parameters, and thus the same form of linear-phase characteristics which 

is given in a compact explicit form in the next expression 

 
( 48 ) ( 48 )1 22 2

1 2
(6, 4, , )

i j
e e

 
 

  
   

  (13) 

In Fig. 4 are shown the amplitude response characteristics of 2D FIR filter of the solution 

from [13] and the proposed solution, respectively, for free parameters 2R=4 and M=6. 



 A Novel Analytical Method for the Selective Multiplierless Linear-phase 2D FIR Filter Function 695 

  

(a)     (b) 

Fig. 4 3D contour plot of amplitude response characteristics, review of the comparison 

between: a) solution from [13], and b) the proposed filter from Eq. (10) 

Fig. 5 shows zoomed forms of pass-bands of filters with attenuation of 0.28 dB of the 

solution from [13] and the proposed solution, respectively, for parameters 2R=4 and M=6. 

    
 (a) (b) 

Fig. 5 2D contour plot of normalized magnitude characteristics, shape of the pass-band 

with attenuation of 0.28 dB; review of the comparison between: a) solution from 

[13], and b) the proposed filter from (10) 

In Fig. 6 are shown the stop-bands of filters with attenuation of 100 dB of the solution 

generated by expressions from [13] and proposed solution, respectively. 



696 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

  

  
 (a) (b) 

Fig. 6 2D contour plot of normalized magnitude characteristics, shape of the stop-band 

with attenuation of 100 dB; review of the comparison between: a) solution from 

[13], and b) the proposed filter from (10) 

In Table 2 and Table 3 are given the values of the surface area of pass-band and stop-

band, respectively, of considered 2D FIR filter function for different values of given 

maximal attenuation compared with 2D FIR filter function given in [13]. Results in Table 

3 are given in (%) in relation to a total area of the amplitude characteristic. 

Table 2 Normalized surface area of pass-band for proposed 2D FIR filter function for given 

values of maximal attenuation compared with 2D FIR filter function given in [13] 

2R M passa (dB) 
Normalized surface area of 

the proposed filter function 

pass-band 

Normalized surface area of the 

filter function pass-band 

proposed in [13] 

4 6 0.28 24.3284219110
-4

 0.6779299310
-4

 

Table 3 Normalized surface area of stop-band for proposed 2D FIR filter function for given 

values of maximal attenuation compared with 2D FIR filter function given in [13] 

2R M stopa (dB) 
Normalized surface area of 

the proposed filter function 

stop-band (%) 

Normalized surface area of the 

filter function stop-band 

proposed in [13] (%) 

4 6 100 21.405425 34.789375 



 A Novel Analytical Method for the Selective Multiplierless Linear-phase 2D FIR Filter Function 697 

6. CONSLUSION 

This paper presents an original approach to the multiplierless linear-phase 2D symmetric 

FIR digital filter function synthesis, bringing the significant improvements in the filter theory. 

The new Christoffel-Darboux formula for classical orthogonal Chebyshev polynomials of the 

first and the second kind is proposed in Appendix of this paper. The presented formula can be 

used for successfully solving extremely complicated problems of the linear-phase 2D filter 

design, with high selectivity and high order. 

Transition from the continuous domain into the 2D z domain is successfully presented. 

This new formula can be directly applied in generating 2D filter functions. All parasitic 

effects, such as Gibbs phenomenon, are suppressed and there is no need for using 

multipliers. Filters design by the proposed method can be applied in various areas, such as 

telecommunications, medicine, pharmacy, seismology, general localizations and diagnostics, 

where they can be of special interest. 

The illustrated examples of the 3D frequency responses and the corresponding 2D contour 

plots of the proposed linear-phase 2D FIR filter are also presented. These examples illustrate 

the high advantages of the proposed approach and an efficient way of designing ultra-selective 

filters. 

The difference between capital research described in [13] and the proposed new classes of 

filter function for even and odd real value of the free parameter 2R, is following.  

Formulae in the z domain proposed in this paper and in the papers [13] are highly 

sophisticated and written on the model of extreme properties of Christoffel-Darboux sum for 

the continuous classical orthogonal polynomials [3]-[5]. 

The proposed multiplierless filter functions do not have the problem of the final 

quantization of filter coefficients, and these solutions are still superior and still very useful for 

real-time and require a minimum area of integrated technology implementation. 

Undesirable Gibbs phenomenon, presented in the analog and in 1D digital filters, has been 

completely eliminated by the proposed solution. In many practical solutions that requires a 

minimum dissipation of DC power supply, this solution successfully realizes circuits without 

multipliers and without quasi multipliers. 

Acknowledgement: The paper is a part of the research done within the project No. 32023, funded 

by the Ministry of Science of the Republic of Serbia.  

REFERENCES 

[1] M. Abramowitz, I. Stegun, Handbook on mathematical function, National Bureau of Standards, Applied 

Mathematics Series, USA, 1964. 

[2] V. D. Pavlović, “New Class of Filter Functions generated directly by the modified Christoffel–Darboux 

formula for classical orthonormal Jacobi polynomials, International Journal of Circuit Theory and 

Applications”, John Wiley & Sons, vol. 40, pp. 1059–1073, 2013. 

[3] S. N. Hazra, M. S. Reddy, “Design of circularly symmetric low–pass two–dimensional FIR digital filters 

using transformation”, IEEE Transactions on Circuits and Systems, vol. 33, no. 10, pp. 1022–1026, 

1986. 

[4] R. H. Yang, Y. C. Lim, “Grid density for design of one- and two-dimensional FIR filters”, Electronics 

Letters, vol. 27, no. 22, pp. 2053–2055, 1991. 

[5] S. H. Low, Y. C. Lim, “Frequency grid density for the design of 2-D FIR filters”, Electronics Letters, 

vol. 32, no. 16, pp. 1460–1461, 1996. 



698 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

[6] A. Klouche-Djedid, S. S. Lawson, “Simple design and realisation of linear phase 2D FIR filters with 

diamond frequency support”, Electron. Letters, vol. 35, no. 14, pp. 1148–1150, 1999. 

[7] N. Vijayakumar, K. M. M. Prabhu, “Two-dimensional FIR compaction filter design”, IEE Proc., Vis. 

Image Process, vol. 148, no. 3, pp. 173–181, 2001. 

[8] V. L. Narayana Murthy, A. Makur, “Design of some 2-D filters through the transformation technique”, 

IEE Proc., Vis. Image Process, vol. 143, no. 3, pp. 184–190, 1996. 

[9] B. G. Mertzios, A. N. Venetsanopoulos, “Fast block implementation of two-dimensional FIR digital 

filters via the Walsh–Hadamard decomposition”, International Journal of Electronics, vol. 68, no. 6, pp. 

991-1004, 1990. 

[10] S. K. Mitra, Digital signal processing. – The McGraw–Hill Companies, New York, USA, 1998. 

[11] S. C. Dutta Roy, “Impulse response of sinc
N
 FIR filters”, IEEE Transactions on Circuits and Systems, 

vol. 53, no. 3, pp. 217–219, 2006. 

[12] D. G. Ćirić, V. D. Pavlović, “Linear Phase Two-Dimensional FIR Digital Filter Functions Generated by 

applying Christoffel-Darboux Formula for Orthonormal Polynomials”, ELEKTRONIKA IR 

ELEKTROTECHNIKA, vol. 4, pp. 39-42, 2012. 

[13] V.D. Pavlović, N. Doncov, D.G. Ćirić, “1D and 2D Economical FIR Filters Generated by Chebyshev 

Polynomials of the First Kind”, Int. Journal of Electr., vol. 100, no. 11, pp. 1592-1619, 2013. 

[14] J.R. Djordjevic-Kozarov, V.D. Pavlovic, “An Analytical Method for the Multiplierless 2D FIR Filter 

Functions and Hilbert Transform in z2 domain”, IEEE Trans. on Circuits and Systems- II: Express 

briefs, vol. 60, no. 8, pp. 527-531, 2013. 

[15] V.D. Pavlovic, J.R. Djordjevic-Kozarov, “Ultra-selective spike multiplierless linear-phase two-

dimensional FIR filter function with full Hilbert transform effect”, IET Circuits Devices Syst., vol. 8, no. 

6, pp. 532–542, 2014. 

APPENDIX 

1. Proposed mathematical background  

If 
1
( )

r
U x


 and 

1
( )

r
U y


 are two sets of the orthogonal Chebyshev polynomials of the 

second kind [5], where x and y are real variables and r is the order of the continuous non-periodical 

polynomials on a finite interval 1 1x    and 1 1y    respectively, with regard to the non-

negative continuous weight functions, 
1
( )w x and 

2
( )w y , defined as  

 
2

1
( ) 1w x x   (A.1) 

and  

 2
2

1
( )

1
w y

y



 (A.2) 

then, for the orthogonal Chebyshev polynomials of the first kind, Tr(y), and the second kind, 

Ur + 1(x), the following relations are valid 

 
1

1 1 1
1

( ) ( ) ( ) 0 ; , 0, 1, 2, 3,
m k

w x U x U x dx m k m k
 



    (A.3) 

and  

 
1

2
1

( ) ( ) ( ) 0 ; , 0, 1, 2, 3,
m k

w y T y T y dy m k m k


    (A.4) 



 A Novel Analytical Method for the Selective Multiplierless Linear-phase 2D FIR Filter Function 699 

For the polynomial, Tr(y), r-th order norm h2(r), is: 

 
1

2

2 2
1

, 0

( ) ( ) ( ( ))
, 1, 2, 3,

2

r

r

h r w y T y dy
r









  




 (A.5) 

and for  the polynomial Um + 1(x), m-th order norm h1(m), is 

 
1

2
 1 1 1

1

( ) ( ) ( ( )) , 1, 2, 3,
2

mh m w x U x dx m





    (A.6) 

A novel analytical method for the linear phase two-dimensional symmetric FIR digital filter 

functions generated by applying the modified Christoffel-Darboux formula with alternating sign. 

Components of that sum are determined by multiplication of orthogonal classical Chebyshev 

polynomials of the first and the second kind, with x and y as a real continual variables, Ur+1(x), 

Ur+1(y) and Tr(x), Tr(y), r = 1,2,...,n (where n is the order of the continual orthogonal polynomials), 

on the equal finite interval [1,1], is proposed in the following explicit form of the orthogonal 

components: 

or 

 
1 1

1
1 2 1 2

( ) ( ) ( ) ( )
( , ) sin ( ) sin ( )

( ) ( ) ( ) ( )

n
r r r r

n
r

T x U x T y U y
x y x y

h r h r h r h r

 



    (A.7) 

i. e. 

 

2

1 1
1

2
( , ) sin ( ) sin ( ) ( ) ( ) ( ) ( )

n

n r r r r
r

x y x y T x U x T y U y


 


 
    

 
 (A.8) 

Using the standard technique, the Eq. (A.8) can be mapped into the new domains, analogue, s, 

and digital, z, [3, 4, 17-19]. Thus, in the z1 domain for example, the following relations are always 

valid: 

 

1 1

1 1 1 1

22
2 2 2 2

( cos ) cos ( ) ( ) / 2 ( ) / 2

( cos ) cos ( ) ( ) / 2 ( ) / 2

j k j k
k k

k

j kj k k k

k

T x k e e z z

T y k e e z z

 



 

 







     

     

 (A.9) 

where Tk (x = cos 1) and Tk (y = cos 2) are the orthogonal continuous Chebyshev polynomials of the 
first kind, and  

 

1 1

1 1 1 1 1

2 2

2 1 2 2 2

sin ( ) ( ) sin ( ) ( ) /(2 ) ( ) /(2 )

sin ( ) ( y ) sin ( ) ( ) /(2 ) ( ) /(2 )

i k i k
k k

k

j k j k
k k

k

U x k e e i z z i

U k e e j z z j

 

 

 

 











    

    

 (A.10) 

where Uk+1 (x = cos 1) and Uk+1 (y = cos 2) are the orthogonal continuous Chebyshev polynomials 
of the second kind.  

As we can see, for odd k, e.g. k = 9, following Eq. (A.12) and Eq. (A.13) the orthogonal 

Chebyshev polynomials Tk (y) and Uk+1 (x), respectively, becomes 



700 J. R. DJORDJEVIĆ-KOZAROV, V. D. PAVLOVIĆ 

 
9 9

2 2 9 9

9 2 2 2
( ) cos ( 9 ) ( ) / 2 ( ) / 2

j j

T y e e z z
 





      (A.11) 

and 

 
9 9

1 1 9 9

1 10 1 1 1
sin ( ) ( ) sin ( 9 ) ( ) / 2 ( ) / 2

i i

U x e e z z
 

 



      (A.12)