In this paper I have established the result on product summability of Borel and Cesaro of first order
1. Difinitions and notation
Definition 1: An infinite series  with the sequence {Sn} of its partial sums is said to be summable (C,1) if (1.1) lim  n ® ¥ i.e. lim sn ® S where sn =  n ® ¥

Definition 2 : An infinite series  with the sequence {Sn} of its partial sums is said to be summable by Borel exponential means or sammable

(B) to a finite number S, if

(1.2) Bp 

Definition 3: An in finite series  with the sequence of its partial sums {Sn} is called (B) (C,1) to a finite number S, if

(1.3) 

where sn stands for the (C,1) transform of Sn is given above.

Let f(x) be a 2p periodic function of x and integrable (L) over the interval

(-p,p). Suppose that Fourier series associated with f (x) is

Then the series

obtained by diff. (1.4) w. r to x is know a the derived Fourier series of f(x) and is not necessarily a Fourier Series

For fixed x ans S, we shall frequently use the following notation
f(x+t) + f(x-t) - 25
y0 = f(x+t) - f(x-t)
g(t) = y0(t)/4sin (t/2)