Question
I am the proud owner of a mature pecan tree which was planted by some previous owner probably several decades ago.  This is the only pecan tree I know of in the area.  It seems that suddenly the grass near and under the tree is doing very poorly, but only in the vicinity of the pecan tree.  An automatic sprinkler system takes care of watering the lawn religiously so it does not dry out and most of the grass gets quite a bit of sun.  Do you have any theories you might be able to throw out to explain what is causing the grass to die?  I believe the grass is a variety of Saint Augustine grass.  The property is located in the Dallas-Fort Worth region.  The pecan tree is very healthy.

Answer
Allelopathic Juglones.

'Allelopathic' is 'inhibition of growth in one species of plants by chemicals produced by another species' (American Heritage Dictionary).

'Juglones' are toxins that shut down respiration in groundcovers and other things that you would normally grow without too much trouble under a Tree.

Fescue and maybe Bluegrass are on the short list
of plants that live long, healthy lives inside the dripline of any Tree that makes Juglones.  But most plant life ceases there; the roots of Pecan Trees ('Carya illinoensis' to botanists) and their relatives -- Black Walnut, Butternut and Hickory -- just OOZE with Juglones.

The jury is still out as to how grass will grow in Juglone-tainted soil.  Some plants wilt, some yellow, some grow in odd patterns.  Some plants are not affected at all.

You need the Grass that Juglone does not bother.  You need Fescue, Kentucky Bluegrass, Orchard Grass, Timothy, Wheat, White Clover or another groundcover that will not find Juglones impossible to live with.  More watering will not change the presence of Juglones.  Sorry to be the bearer of such bleak news.