eng Ansari Education and Research Society Journal of Ultra Scientist of Physical Sciences August 2010 22 2 551 556 jusps-A 1005 article K.S. Bhatia 1 Department of Mathematics, Govt. Science College, Rewa - 486001 (INDIA) <p>In this paper I have established the result on product summability of Borel and Cesaro of first order<br /> 1. Difinitions and notation<br /> Definition 1: An infinite series &nbsp;with the sequence {Sn} of its partial sums is said to be summable (C,1) if (1.1) lim &nbsp;n &reg; &yen; i.e. lim sn &reg; S where sn = &nbsp;n &reg; &yen;</p> <p>Definition 2 : An infinite series &nbsp;with the sequence {Sn} of its partial sums is said to be summable by Borel exponential means or sammable</p> <p>(B) to a finite number S, if</p> <p>(1.2) Bp&nbsp;</p> <p>Definition 3: An in finite series &nbsp;with the sequence of its partial sums {Sn} is called (B) (C,1) to a finite number S, if</p> <p>(1.3)&nbsp;</p> <p>where sn stands for the (C,1) transform of Sn is given above.</p> <p>Let f(x) be a 2p periodic function of x and integrable (L) over the interval</p> <p>(-p,p). Suppose that Fourier series associated with f (x) is</p> <p>Then the series</p> <p>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</p> <p>For fixed x ans S, we shall frequently use the following notation<br /> f(x+t) + f(x-t) - 25<br /> y0 = f(x+t) - f(x-t)<br /> g(t) = y0(t)/4sin (t/2)</p> https://www.ultrascientist.org/paper/1005/ Summability derived fourier series by (B)